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SUPERPROCESSES By HRVOJE SIKIC A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993 ACKNOWLEDGEMENTS Grateful acknowledgements are due to persons whose help and support made this work possible. First, and foremost, my advisor Prof. Murali Rao is thanked for his guidance throughout my study. His original insight into mathematics and his humanity are a constant inspiration for me. Prof. Zoran R. PopStojanovi6 already directed some of my work in Zagreb. His vast knowledge made mathematical discussions particularly enjoyable. Prof. Joseph Glover helped me very efficiently on numerous occasions, ranging from research questions to organizational problems. I also wish to thank Prof. James K. Brooks and Prof. Andrew J. Rosalsky for their service on my committee. I wish to thank faculty members of the Department of Mathematics of the Uni versity of Zagreb for giving me a strong mathematical foundation. In particular, I am grateful to Prof. Nikola Sarapa, who introduced me to probability and led me to magister degree, and to my longtime friend and colleague Dr. Zoran Vondracek, with whom I worked from our first mathematical steps until today. And last, but not least, I wish to express my gratitude to the members of my family. My parents, whose working attitude and understanding of criticism as the basic approach to science, are lasting examples for me. My wife Sonja, who, be ing a mathematician herself, was a supportive and understanding companion in the sometimes very demanding life of a mathematician. TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT ........................ CHAPTERS 1 INTRODUCTION ............................... 2 GENERAL RESULTS ..... ....................... 2.1 Nonlinear Perturbations of Positive Semigroups 2.2 Supersemigroups and Superprocesses . 2.3 Kac Semigroup and Linear Case . . 2.4 Feller Condition and Infinitesimal Generator . 2.5 Branching Property. . .. 3 ONEDIMENSIONAL CASE . . 3.1 Supersemigroups and Infinitesimal Generators . 3.2 Time Change and ZeroOne Law . 3.3 Measures on Rays ................. 3.4 (Idt, IF )Supersemigroups . . 4 TRANSITION PROBABILITIES OF SUPERPROCESSES ........ 4.1 TrotterKato Formula ........................... 4.2 Superprocesses over Markov Chains ... ................. 4.3 Superprocesses over Deterministic Markov Processes .......... 5 CONCLUSION ................... .............. REFERENCES ................................... BIOGRAPHICAL SKETCH ............................ . . iv 88 88 95 100 107 112 116 11111 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SUPERPROCESSES By HRVOJE SIKIC May 1993 Chairman: Dr. Murali Rao Major Department: Mathematics DawsonWatanabe superprocesses (Xt), over a Markov process (t), are usually defined by transition supersemigroups Qt(p, dv). The Laplace transform of Qt is determined by a nonlinear semigroup (V,), which is obtained as the perturbation of Markov semigroup (Tt) (given by () by a branching mechanism T. In the first part of this work we establish a general analytical framework for nonlinear perturbations of positive semigroups, which preserve positivity. The semi groups involved are only weakly measurable with respect to the subspace M of the space of linear functionals. The wellknown results on superprocesses by Fitzsimmons and Watanabe are obtained as special cases. In this fashion (Xt) can be defined as Mvalued process, in general. Very useful and new, Fubinitype lemma is applied several times in this part; among other things to estimate (Vt) from below and above in terms of the Kac semigroup of (Tt). In the second part we analyze (Xt) through (Vt) and (Qt). We start by detailed account of onedimensional superprocesses (studied by Lamperti in the late sixties). Some new formulas for (Vt) and (Qt) are derived. These results are then used to give iv detailed description of "component processes" (XT) (which depends only on ') and (X4) (which depends only on 1). The new notion of a measure on rays is introduced to obtain the description of (Xt"). It is shown that (Qt) can be approximated, by TrotterKato type formula, using (Xt) and (X\). Some explicit examples are given. It is shown that in the case of Markov chain (), the local behaviour of (Xt) can be obtained, by means of CameronMartin formula, from (X*). Finally, for deterministic ((t) the explicit formulas for (Xe) are obtained, which show that "the mass" of (Xt) is governed by T, while the "spatial movement" of (Xt) is governed by (&). In section 3.2 a ZeroOne law for Brownian motion is obtained, which complements EngelbertSchmidt ZeroOne law. CHAPTER 1 INTRODUCTION Discretevalued branching processes were studied in the 19th century as math ematical models in a discussion of the survival of family names. In this century applications broadened toward studies of the survival of mutant genes, nuclear chain reactions, and many others. Considering the problem of population growth, it was suggested by William Feller that in the case of large populations continuous models work better than discrete ones (which are successful in the case of relatively small populations). He presented his programme to the mathematical public in 1951 [26]. In this paper Feller introduced a class of diffusion processes, with values in the set of positive real numbers, which can be obtained as limits of discretevalued branching processes. Feller also computed the transition probabilities of these diffusions and characterized them in terms of corresponding differential equations. The consequences of Feller's work [26] are extensive, and we will not attempt to follow all of them. However, two lines of developments coming from Feller's work [26] are important to us. We will see that this two lines later merged. The first of these two lines, and rather central to us (although it appeared later than the second one), started with the pioneering work [8] of Donald Dawson in 1975. Dawson followed the line of extension of Feller's consideration to the space time context. Namely, although Feller's diffusions are continuum valued, they are models for the number of individuals (or total mass) of a population. The idea was later to find models for populations distributed in space, which are changing through time. Such processes will have to be measurevalued, and, as it was already noticed before 1975, will be described by nonlinear stochastic evolution equation. Dawson's idea was to consider n individuals of equal mass located at n points yi,..., y, in the space E C Rd. Each individual has an exponentially distributed lifetime with mean 1/7, and at the end of its lifetime it is replaced by r similar offspring with probability pr, all initially located at the final position of their parent. Each individual moves independently through space E, according to the given law of a Markov process (.t), whose infinitesimal generator we denote by A. If N(t) is the total number of individuals at the moment t, then the mathematical model for the situation above is the branching diffusion N(t,w) X"(w) = E 6,(w) (1.1) i=I where S6 is the pointmass at y E E. The continuous analog of (Xn/) led Dawson to the stochastic evolution equation dX(t) = (A + a)X(t)dt + 2yX(t)dB(t) (1.2) where B(t) is, so called, Brownian motion with spatially correlated increments. The main technical difficulty, in giving a mathematical meaning to equation 1.2, is the appearance of the nonLipschitz factor 2X(t). By treating the characteristic func tionals of (Xf) through the application of semigroup theory, Dawson succeed in defining rigorously measurevalued process (Xt),. which can be understood as the so lution of 1.2. He also showed that, by properly adjusting the parameters, (Xt) is the limiting case of branching diffusions (Xy) as n + +oo. It was done in the case when (6() is a standard Brownian motion, and po = 1/2 2a/n, P2 = 1/2 + 2a/n. The process (Xt) is the example of what was later to be named superprocess. In doing so Dawson not only offered one of the main ideas for construction of superpro cesses, but also gave the intuitive meaning to (Xt) as the mathematical model for the propagation of a group of individuals (or cloud of particles, for example) through space and time. In the succeeding few years Dawson outlined main directions in the study of multiplicative (i.e. branching) stochastic measure diffusion processes, as he called them then, or superprocesses, as most of the authors call them today. His programme was carried through most of the eighties. At the end of the eighties and the beginning of the nineties several new ideas emerged, but we will say more about it later. Except for the paper [8], two other papers in the late seventies were crucial for the development of the theory of superprocesses. In 1977 Dawson, in his paper [9], studied the limiting behaviour of superprocesses, i.e., the behaviour of Xt as t ) +oo. In particular, the questions of extinction and of the existence of invariant distribution were studied. This line of study was continued and developed [11], [19], [32]. I. Iscoe [34] introduced weighted occupation time processes and investigated their behaviour when t  +oo. Another important early paper was published in 1979 by Dawson and Hochberg [12]. In this paper local properties were studied, i.e., if we fix t > 0, then we can ask questions about the nature of the measure Xt. In many cases it is a singular measure, so the questions about the support of Xt naturally came into consideration. This type of questions were further studied by Sylvie RoellyCoppoletta in her work [52] and other works [58] and [43]. In 1988 Edwin Perkins [48] considered the question of the nature of the random measure over time intervals (i.e. not just for a fixed t > 0). Special consideration was given to the case of the superbrownian motion (superprocess over standard Brownian motion (at)), and to the study of its paths [13]. All this led to the development of an important new idea, i.e., the idea of a historical process. It is a superprocess with values in the space of measures on the space D(E) of Evalued cadlag paths on R+ = [0, +oo). Intuitively, the historical process takes into consideration the "genealogy" of the starting superprocess (Xt). For detailed study we refer to Dawson and Perkins [15]. Let us also mention the paper [25] by Steven Evans and E. Perkins in which they introduce the notion of a collision measure. They use it to "gauge" the overlap of a fixed measure and a superprocess. Throughout all that period one of the main problems was the problem of the construction of superprocesses. Dawson's starting idea, given in 1975, was later for malized in terms of a martingale problem (developed by Stroock and Varadhan). We refer to [14], [52] for these results, and to [24] for general treatment. In [52] the mar tingale properties of superprocesses were emphasized. In the following years more general processes (6t), more general conditions on branching {p,}, and more general initial conditions were considered. For construction of superprocesses on ptempered distributions see [34], for further formalization in terms of stochastic differential equa tions see [43], and, together with further study of martingale properties, [46]. Already in the eighties superprocesses (Xt) were characterized as measurevalued processes, such that their Laplace transforms were given by E [e f )x] = e(vf)( ; X0o = (1.3) where f > 0 is bounded, continuous function on E, and (Vt) is a nonlinear semigroup which satisfies the equation Vtf = Ttf + j T,t[,(V,f)]ds (1.4) where (Tt) is the Markov semigroup of (6t) and T(A) = y2. These formulas sug gested strong connections with the work on branching processes done by S. Watanabe in 1968. It was further exploited by P. Fitzsimmons and E. Dynkin. But before we go to their work, let us exploit the other line of development related to Feller's work mentioned at the beginning. In 1951, when Feller offered his programme, the notion of branching processes meant only the discretevalued processes. Miloslav Jifina, in his paper 1958 [36], developed the notion of branching property to processes with continuous state space. Since additive structure of state space is needed, measurevalued processes rep resent an important example. However, it was not known how to characterize continuousvalued processes as branching. Of course, the simplest case was the case of R+ = [0, +oo). Realvalued branching processes were characterized as timechanged Levy processes by John Lamperti in 1967 [44]. It turned out that R+valued branch ing diffusions are exactly the diffusions obtained by Feller in 1951. The next step, i.e., the characterization of R+valued branching processes was carried out by Shinzo Watanabe in 1969 [57]. To simplify notation, he worked on the case n = 2. Watan abe characterized completely these processes in terms of their infinitesimal generators, and, in the case of diffusions, analyzed the behaviour on the boundary of R'. Inde pendently, same results for infinitesimal generators were obtained by Yu.M. Ryzhov and A.V. Skorokhod in 1970 [51]. More general branching processes with immigra tion were treated in another work [40]. An account of branching semigroups on the space of finite measures on compact separable metric space was given in 1969 by M.L. Silverstein [55]. Most recently, Dynkin, Kuznetsov, and Skorokhod studied the structure of general branching measurevalued processes [22]. However, the most im portant work for our consideration is the paper [56] by S. Watanabe, published in 1968. Watanabe constructed the class of measurevalued branching processes (Xt), characterized by 1.3 and 1.4, where E is a separable compact metric space, and T is, so called branching mechanism, given by S(A) = ao + a A a2A2 eAu 1 + n(du) (1.5) Jo0 1 + u ) where A E (0, +oo), ao, al, a2 E R, ao, a2 > 0, and n is a measure on (0, +oo) such that 0(0 U du)< +oo (1.6) His assumption was that (,) is a Feller process, and then he proved that (Xt) is also Feller and characterized its infinitesimal generator. Now we can go back to the story of superprocesses. In 1988 Patrick Fitzsimmons offered a new way of constructing superprocesses, following Watanabe's approach. He considered a very general class of Markov processes (st), and very general form of xF with nonconstant coefficients. He showed [28], [29] that the corresponding, i.e., ((, I) superprocess can be constructed and that it satisfies nice regularity properties. The superprocess is now defined so that it satisfies 1.3 and 1.4. Similar approach was taken by Eugene B. Dynkin [18]. He considered branching mechanism T(A) = A2, and time nonhomogeneous case. Also, his methods are different from Fitzsimmons'. In this way Watanabe's and Dawson's theories were formally unified. Dynkin is also re sponsible for the introduction of the term superprocess over Markov process (t) with branching mechanism I'. We will follow this terminology and mostly Fitzsimmons' line of approach in this work. We will actually use the term DawsonWatanabe super process, since there are other types of processes (FlemingViot, OrnsteinUhlenbeck) sometimes called superprocesses, which will not be considered in this work. Let us finish this brief survey of the theory of superprocesses by mentioning some other recent developments, which otherwise will not be treated in this work. The third, very different, construction of the superbrownian motion was recently offered by JeanFrancois Le Gall [30]. The connections between the theory of superprocesses and PDEs has been investigated by Dynkin. The systematic treatment of these problems is given in [20]. Part III of this paper contains a nice historical survey of the theory of superprocesses. The LUvyHincin representation for superprocesses is analysed in [39]. Despite all these achievements and thorough analysis of some particular cases, like superbrownian motion, it appears that the "nature" of superprocesses is still somewhat hidden from us. For example, even in the case of Markov chain (&t), where the corresponding superprocess (Xt) is an R"valued branching process, and we can characterize the infinitesimal generator of (Xt) completely (see [57]), it is still not completely clear which part of the behaviour of (Xt) comes from (&t) and which from 'k. Intuition would suggest that (it) influences "spatial motion," while V governes "changing of the mass." Can we describe mechanisms which govern these behaviours? And what about other type of processes, in more general spaces? It seems that one of the main difficulties is in the fact that equation 1.4 is nonlinear, and in general we can not expect to have explicit solutions of 1.4. However, if we concentrate only on "components" of superprocesses we can hope to characterize, through explicit formulas even, these "components" rather completely, and then to use them to recover some information from the most general cases. After establishing superprocesses in a very general situation, we will follow this line of approach, i.e., we will try to give as explicit formulas for the "components" as we can, and then to recover general formulas from these, if possible. We will say what we can in general cases, but we will concentrate on the simplest examples of ((i). Although we can not completely answer the questions mentioned above, we hope that this work offers some improvement and clarification in this direction. The thesis is organized into five chapters, where the first chapter is this intro duction, and the fifth chapter is a conclusion. Hence, the results are contained in chapters 2,3, and 4. In chapter 2 we deal with the problem of the foundation of superprocesses. We take an approach through the semigroup theory of Markov processes. Hence, we have to solve a nonlinear integral equation first, and then to show that the result is the exponent of the Laplace transform of the transition probability of the superprocess. This method was used by both Watanabe and Fitzsimmons, but the first dealt with the Feller semigroup on a compact state space, while the second dealt with more general Lusin state space. In section 2.1 we offer a unified approach, so that both Watanabe's and Fitzsimmons' result are special cases of our general method, as is shown in sections 2.2 and 2.4. Actually our method solves the problem of positivity preserving nonlinear perturbation of positive semigroups, and the class of semigroups included is very large. Particularly useful is Lemma 2.1.3, which we will apply to deal with linear case and Kac semigroup in section 2.3. In section 2.5 we mention the branching property of superprocesses, and how it simplifies our dealing with the transition probabilities of superprocesses. In chapters 3 and 4 we analyze the transition probability for particular superpro cesses. We start by describing the simplest, onedimensional superprocesses, which were studied first by Lamperti. We compute some new explicit formulas in these cases. In section 3.2 we also prove a zeroone law for Brownian motion, which com plements the EngelbertSchmidt zeroone law. In section 3.3 we introduce a notion of measures on rays; these are example of measures on the space of measures. Using this notation we can precisely describe transition probabilities for superprocesses over (, where t = Jo, denoted by (X'X). This will be done in section 3.4. In chapter 4 we use results from the previous chapter to interpret superprocesses in terms of their "component processes" X" and X, which are much simpler. X is obtained when IF = 0 and is a deterministic process, which is easy to describe. In the 9 general case we can only approximate the transition probability of X through tran sition probabilities of the "component processes" (see section 4.1). However, in some special cases we can have more explicit interpretations. The case of deterministic (t) is rather completely resolved in section 4.3, where we obtain nice explicit formulas. The case of Markov chains is already more complicated and we were able only to describe the local behaviour of X through the local behaviour of X*, by application of CameronMartin formula. This will be shown in section 4.2. CHAPTER 2 GENERAL RESULTS 2.1 Nonlinear Perturbations of Positive Semigroups In this section we consider the problem of nonlinear perturbation of a linear posi tive semigroup, in such a way that the positivity of semigroup is preserved. Although our approach relys on standard iteration method and is specially suited for our pur poses, we believe that the fact that we consider a very general class of semigroups and nonlinear operators, makes it important even in an abstract framework of math ematical analysis. A good reference for the standard case, i.e., linear perturbation of not necessarily positive and Cosemigroups is [47], and for the case of perturbation by branching mechanism (see the next section) the references are [56], [28], and [18]. Let (B, II <) be a Banach space and a linear lattice with closed positive cone. More precisely, (B, II ) is a Banach space and (B, <) is a partially ordered set with the following properties: for every x,y,z E B x < y == x + z < y + z (2.1) for every x,y E B and real A > 0. x < y == Ax < Ay (2.2) for every X B there exists sup{z, 0}, denoted by x V 0 or xz (2.3) B = {x E B I x > 0 } is closed in the norm topology (2.4) It can be shown (see for example [53]) that in (B, <) x V y and x A y exist, for every x,y E B, and that xVy + xAy = x+y. Then we define x to be () V0, and zIx to be x+ + x. Some of the properties that x+, x, and lzx satisfy are: S= x+ x, i.e.,B=B+ B+ (2.5) 10 x+ v x = xV (x) = xI X+ A x = 0 (2.6) Ix+yi y < xl + yl I = I XI Ij (2.7) Remark 2.1.1 Notice that (B, 11 I, <), as defined above, is not necessarily a Banach lattice. Banach lattice is a partially ordered Banach space which satisfies 2.12.3, and the property that, for every z,y E B, I1 _< Yl == \\lx < Ilyll (2.8) If B is a Banach lattice then the mappings x ,* x+, x x, (x,y) x V y are uniformly continuous, and 2.4 is satisfied. See [53], 11.5. for details. Therefore, the class of Banach spaces that we consider is bigger than the class of Banach lattices. Typical example of the space which satisfies 2.12.4 is a Sobolev space H'((a, b)) with the relation of partial order defined by ( f < g if f(x) < g(x) for every x E (a, b)). But H'((a, b)) is not a Banach lattice. It is easy to construct a sequence of nonnegative functions {f,} in Hl((a,b)), such that 0 < f, < 1 and lf,\llL2 +oo. Thus H'((a, b)) does not satisfy 2.8.0 As usual, we will say that an operator (not necessarily linear) O : B B is positive if O(B+) C B+. Let B* be the dual space of B, i.e., the space of all bounded linear functionals on B. We define the cone of positive functionals B*,+ by B*+ = {f E B* I f(x) >0, for every x E B+ (2.9) It is true that, if f(x) > 0, for every f E B*'', then x C B+. This statement follows from the fact that B+ is closed, see [41] p.226. We will slightly generalize the notions of weak measurability and weak integration (see the examples below for the cases which require such generalizations). Let M C B* be a nonempty subset of B* with the following properties: (M,  II) is a Banach space, where 11 is the operator norm of B* (2.10) m(x) = m(y), for every m 6 M ==Z x= y (2.11) m(x)> 0, for every m M = M B*'+ x E B+ (2.12) 1 =sup ; meM \ {0} (2.13) Notice that, if M = B*, then M satisfies 2.102.13. We define aalgebra M on M to be the smallest aalgebra such that, for every x E B, the mappings m m(x) are measurable. Let I C R be an interval and B(I) the aalgebra of Borel sets on I. Definition 2.1.1 A vectorvalued function x(s) : I  B is: Mweakly measurable ifm o x is B(I)/B(R) measurable, for every m E M. Mmeasurable if (m, s) i m[x(s)] is M ( B(I)/B(R) measurable. MPettis integrable if there exists an element xj E B, such that, for every m E M, mo x is Lebesgue integrable and m(xi) = m[x(s)]ds (2.14) In this case xj is called MPettis integral of x(s) and is denoted by xi = (MP) f x(s)ds (2.15) If M = B*, then these are the standard notions of weak measurability and Pettis integrability (see, for example, Chapter III in [33]), and in this case we will denote Pettis integral x, by (P) j x(s)ds (2.16) The notion of Mmeasurability requires additional explanation in some cases (see Proposition 2.1.1 and examples below). Definition 2.1.2 A linear bounded operator B : B  B is Minvariant if, for every mE M, moBE M. The basic properties of MPettis integrability are given in the following theorem. Notice that I(s)]I is not necessarily measurable, even if x(s) is MPettis integrable. Various formulations of Fubini theorem are possible. We will state the formulation which serves our purposes. Theorem 2.1.1 Let x(s) and y(s) be MPettis integrable on I, and a, f real numbers. Then the following is true: The M Pettis integral of x(s) is unique. (2.17) If, for every s I, x(s) E B+, == (MP) x(s)ds c B+. (2.18) If M1 C M and Mx satisfies 2.102.13, then x(s) is M1Pettis integrable and two integrals coincide. In particular, if x(s) is Bochner or Pettis integrable, then x(s) is MPettis integrable and (B) x(s)ds = (P) x(s)ds = (MP) x(s)ds (2.19) where the first integral is Bochner integral. ax(s) + py(s) is M Pettis integrable and the integral is linear. (2.20) If a(s) : I  [0, +oo) is Lebesgue integrable and, for every s E I, I x(s) II a(s), then (MP) x(s)ds i a(s)ds (2.21) If B is Minvariant, then B[x(s)] is MPettis integrable and B [(MP) x(s)ds = (MP) B[x(s)]ds (2.22) If x(s,t) : I x I  B is such that s * z(s,t) and t I* z(s,t) are MPettis inte grable, and, for every m E M, m[x(s, t)] is B(I) 0 B(I) measurable and integrable on I x I, and t i* (MP) f, z(s,t)ds is MPettis integrable, then s i (MP) f, z(s,t)dt is MPettis integrable and (MP) dt (MP) ds r(s, t) = (MP) ds (MP)f dt x(s,t) (2.23) Proof. The statement 2.17 follows from 2.11, 2.18 from 2.12, and 2.19 by defi nition and the fact that every Bochner integrable function is Pettis integrable and two integrals coincide (see [33], p.80). The statement 2.20 follows directly from the definition, and 2.21 from the definition and 2.13. The statement 2.22 follows from the definition and the fact that, for every m E M, m o B E M and (since x(s) is MPettis integrable) (m o B) ((MP)J x(s)ds) = (mo B)(x(s))ds To prove 2.23 notice that there exists a vector xo E B such that m(Xo) = di m [(MP)Jr ds x(s,t) = J dt J ds m[x(s,t)] Recall that m[x(s, t)] is integrable on I x I and realvalued. Thus, by Fubini theorem for realvalued functions, m(Xo) = ds I dt m[x(s,t)] = ds m [(MP)r dtx(s,t) , where the second equality follows from MPettis integrability of t 4 xz(s, t). No tice that s F+ f dtm[x(s,t)] = m[(MP) f z(s,t)dt] is Lebesgue integrable and the integral is equal to m(xo), for every m E M. By definition, it means that s i* (MP) fJ z(s, t)dt is MPettis integrable and Xo = (MP) ds (MP) dt (s, t) *//0 Notice that the consequence of 2.18 and 2.20 is that MPettis integral is monotone, i.e., if x(s) and y(s) are MPettis integrable, and, for every s E I, x(s) < y(s), then (MP) x(s)ds < (MPJ y(s)ds Q.E.D. In the following text we will consider only operators which preserve Mmeasurabi lity. An operator O : 1(0) C B ) B (not necessarily linear) is M measurable if t O[x(t)] is Mmeasurable for every Mmeasurable function t i* (t) with values in D(0). Lemma 2.1.1 Every Minvariant operator B : B  B is Mmeasurable. Proof. Let x(t) : I B be Mmeasurable. It means that the mapping F(t,m) : I x M R, F(t,m) = m[x(t)] is B(I) (9 M/B(R) measurable. Consider a mapping H(t, m) : I x M I x M defined by H(t, m) = (t, m o B). Notice that H(t, m) E I x M, since B is Minvariant. We claim that H is B(I) 0 M4/B(I) 0 M measurable. It is enough to prove that (t, m) 1 m o B is B(I) 0 M/IM measurable, i.e., it is enough to prove that m F, mo B is MIM measurable. By the definition of M, it is enough to prove that, for every x E B, m ', (m o B)(x) is M/1B(R) measurable. Since Bx E B, it is obviously true. Now, (t, m) '* m[B(x(t))] = F o H finishes the proof. Q.E.D. Let us specify the class of semigroups that we will consider in this section. Definition 2.1.3 A semigroup (Tt; t > 0), To = id, of bounded linear operators on B is a positive, MPettis integrable semigroup if it satisfies the following properties: Tt is M invariant and positive, for every t > 0 . (2.24) There exist D > 1 and w > 0, such that liTtil < D e", for every t > 0. (2.25) For every x(t) : (0, +oo) B, Mmeasurable and bounded on bounded subinter vals, the mapping (m, s, t) ' m[T.[x(t)]] is M 0 B((0, +oo)) 0 B((0, +oo))/B(L) measurable and, on every bounded subinterval, the mapping s i T,[x(s)] is M Pettis integrable. (2.26) Conditions 2.24 and 2.25 are standard. Condition 2.26, which is rather technical, enables us to use Fubinitype theorem and weak integration. We will see in examples that 2.26 is fulfilled under more standard assumptions. Example 2.1.1 Let (B, I[ 1, <) be a separable Banach space and a linear lattice with closed positive cone. Among such spaces are LP(I), 1 < p < +oo, where 1L is a afinite measure defined on a countably generated aalgebra, Co(X), where X is a locally compact Hausdorff space with a countable base, and Sobolev spaces W"P(I), where 1 < p < +00. In all these spaces < is inherited from the order on the real line. Let M = B*. Notice that in this case every bounded linear operator is Minvariant. Let (TI; t > 0) be any positive semigroup of bounded.linear operators, which satisfies 2.25 and such that t Ttx is weakly measurable. Since B is separable, t " Ttx is strongly measurable, and, since (Tt) is a semigroup, t (0, +oo) ' Ttz is strongly continuous (see [33], Theorem 10.2.3.). Consider now a mapping (s, f) (0, +oo) x B*  B* given by (s, f) f o T,. For every x E B, (s, f) f(Tx) is Mmeasurable in the second coordinate, and continuous in the first coordinate s G (0, +oo). Since (0, +oo) is a separable metric space, (s,f) } f(Tbx) is B((O, +oo)) (9 M/B(R) measurable. Therefore (s, f) F fo T, is B((0, +oo)) M/M measurable. If x(t) is Mmeasurable and bounded on bounded subintervals, then (f, s, t) (f o T,,t) (f o T)(x(t)) is M (0 B((0, +oo)) 0 B((0, +oo))/B(R) measurable. In particular, (f, s) ( f (f o T,)(x(s)) is M 0 ((0, oo))/B(R) measurable. Since M = B*, s * T,[x(s)] is weakly measurable, and s jjIT[x(s)] is bounded on bounded subintervals. B is separable implies that s * T,[x(s)] is strongly measurable, and, therefore, Bochner integrable on bounded subintervals. It shows that (Tt) satisfies 2.26.0 Example 2.1.2 Let (E, ) be a measurable space, and (B, I1 ) a Banach space of bounded Emeasurable realvalued functions with the sup norm. Then we have a natural partial order on B, inherited from the real line. Such a space B satisfies 2.12.4, and, even more 2.8. Let M be a set of all finite signed measures on (E, E). Then M with the total variation norm satisfies 2.102.13, and M 7 B*. Let pt(x, dy) be a positive kernel on E x such that (t, x) pt(x, dy) is measurable, and there exist D > 1, w > 0, such that, for every t > 0 and x E E, pt(x, E) < D exp(wt). We define Tt on B by (Ttf)(x) = f(y)pt(x,dy) (2.27) Typical example of such a semigroup is a Markov transition function pt(x, dy) and its semigroup, when it is a transition function of a Borel right process with the state space (E, B(E)), where E is a Lusin topological space and B(E) the aalgebra of Borel sets (see [54], pp.13, 105, and App.2). In this case w = 0 and D = 1. Let us show that (Tt), defined by 2.27, satisfies all the requirements of Definition 2.1.3. Since pt(x,dy) is a positive kernel and IITt l = sup{pt(x,E);x E E}, 2.24 and 2.25 follows immediately from our assumptions. Let us denote JE f(x)1.(dx) by (f, p), where f E B and y, E M. Consider a function F(s,x): (0, +oo) x E  R defined by F(s,x) = 1B(s) 1H(), where B 6 B((0,+oo)), H E Then (F(s,.),A) = 1B(s) j(H), and, therefore, (s, A) '* (F(s, ), A) is B((0, +oo)) 0 M/L/(;.) measur able. Using the monotone class argument, it follows that (s,' ) (F(s,.), ) is B((0, +oo)) 0 M/3B() measurable for every bounded B((0, +oo)) /B(R) measur able F(s, x). By our assumption, F(s, x) = (Tf)(x) is such a function, for every f E B. Hence, for every H E E and B E B((0, +oo)), (1B(t) 1H('), AT.) = lB(t) (T.(1H), /) which implies that (IL, t, s) '4 (G(t, ), IT,) is jointly measurable, for G(t, y) = 1H(y)* 1B(t). By monotone class argument, the same is true for any jointly measurable and bounded function G(t,y). Notice that we can identify (E, ) with ({6,; x E E}, M n {6,; x E}), where 6, is a point mass at x. Therefore, if f(s) : (0, +oo)  B is M measurable, then (s, x) F f(s)(x) = (f(s), ,) is B((0, +oo)) /B(lR) measurable and bounded on any bounded subinterval. It follows that (I, s, t) F (T,[f (t)], J) (f (t), JT,) is M ( B((0, +oo)) 0 B((0, +oo))/B(R) measurable. In particular, it shows that (s, x) f T,[f(s)](x) is B((0, +oo)) 0 EI/B(IR) measurable and bounded on bounded subintervals. It follows, by Fubini theorem, that fo() = f,)T,[f(s)](x)ds exists, for every x E E, and defines a bounded Emeasurable function fo. By Fubini theorem, for every p E M, (fo, ) = ,)(T, [f(s)], ) ds which shows that s H T,[f(s)] is MPettis integrable on every interval (0, t). Hence, (Tt) satisfies all the requirements of Definition 2.1.3.0 The main problem of this section is the perturbation of a positive, MPettis in tegrable (linear) semigroup. Let (Tt) be a linear, positive, MPettis integrable semi group (as defined in Definition 2.1.3). Let L : B+ B be an Mmeasurable (not necessarily linear) operator such that L[x(t)] is bounded on bounded subintervals, whenever x(t) E B+ is bounded on bounded subintervals. We would like to solve an integral equation Vtx = Ttx + (MP) (o T,[L(Vtx)]ds (2.28) where x E B+. When we say that (Vt) is the solution of 2.28, we mean that (Vt; t > 0) is a family of operators Vt : B+ B+, such that t 4 Vtx; is Mmeasurable and bounded on bounded subintervals, and (Vt) satisfies 2.28. Notice that under these assumptions s '* T,[L(Vt_,x)] is MPettis integrable (see 2.26). Thus, 2.28 makes sense. We will solve the problem by iteration method, which relies on wellknown Gron wall's inequality. In the case of superprocesses the method has been used in the works of Watanabe [56], Fitzsimmons [28], and Dynkin [18]. Gronwall's inequality states that if a : [0, +oo) [0, +oo) is a bounded measurable function with the property that, for every t > 0, a(t) a+ b a(s)ds a,b> (2.29) then, for every t > 0, a(t)< a ebt (2.30) In particular, if a = 0 then a = 0. The problem is that some of the functions that we will consider are not measur able. However, we can easily adjust the standard proof of the Gronwall's inequality, to obtain the following, say nonmeasurable, version of the Gronwall's inequality. Lemma 2.1.2 Let a : (0,)  [0, c] be a bounded mapping, where 0 < r < +oo, 0 < c < +oo. If there is a family of measurable mappings {/31; 7 E t E (0, )}, (0 : (0,t) [0, +oo), such that, for every t e (0, 7), and, for every s (0, t), ft a(t) < a + sup Pt(s)ds (s) < b a(s) , (2.31) (2.32) where a > 0, b > 0 are constants, then, for every t E (0, r), a(t) < a e b (2.33) Proof. Notice that a(t) is bounded, but is not necessarily measurable. We claim that, for every n E N, ( 1 (bt) (bt)" a(t)_ l  + cn , k=0 and 2.33 follows then directly. a(t) < a + sup /f t(si)ds, 1i 0 (2.34) 3 i, (s) < b .a(si) ab + sup bf (s,)ds, I I I  (sn1) < b a(si) < ab + sup b ) Tn Hence, .(,Sn1) < ab + fn" b2cdsn, and, going backwards with these inequalities, we obtain a(t) < a + abds, + ab2ds2ds +...+ Jo Jo Jo + /' ". o J0 b"c ds ... ds , which implies 2.34. Q.E.D. Let us give sufficient conditions on L, such that the solution of 2.28 exists, is bounded and unique, and has a semigroup property. We will start with boundedness. Theorem 2.1.2 (Boundedness) If (Vt; t 0) is a solution of 2.28 and L satisfies one of the following two conditions: a) there exists the smallest nonnegative number, say IIL II, such that, for every x E B+, IILxzl < IILI l\x\ (2.35) b) B is a Banach lattice, x i4 x+ is Mmeasurable operator, and there exists the smallest nonnegative number, say IIL+II, such that, for every x E B+, II(Lx)+Il < IIL+l Il x (2.36) then there is a constant K > 0, such that, for every x E B+ and t > 0, IIVfxll < De"t lxl (2.37) In the case a) K = w + IILID, and in the case b) K = w + IIL+IID, where w and D are defined by 2.25. Proof. Let us prove the case b). The case a) is analogous, but simpler than the case b). Notice first, that, since 2x+I = xz < x + x = I, the operator x '* (Lx)+ is Mmeasurable and (L[z(t)])+ is bounded on bounded subintervals, whenever x(t) E B+ is bounded on bounded subintervals. Since (Vt) is the solution of 2.28 it follows that s T,[(L(Vt_,x))+] and s Tt_,[(L(VSx))+] are MPettis integrable and that, for every x E B+, 0 < Vtx = Ttx + (MP) j(,t Tt.[L(V)x)ds Tt_.[(L(Vz))+]ds (MP) 0Ot) Since, for every s E (0, t), Tt_, is positive, we obtain 0 < Vtx < Tt + (MP) 0) T,_[(L(V.a))+]ds B is a Banach lattice, thus, by 2.8, we obtain IIVt I < IITtxI + (MP) (o,t) (2.38) Let 0 < T < +oo be fixed. By assumption t E (0,r) i IIVtxll is bounded. Let a(t) = exp(wt)1lVtxll Then a : (0,r)  [0,+00) is bounded, and, by 2.25 and 2.38, a(t) < DIIxll + sup mEM\{0} < Djllj + m {(MP) f(o,t Tt, [(L(Vx))+]ds}\ im Ie" supM\ et J m(Tt[(L(Vax))+])d mMEM\o} 01m1 Therefore, we choose S( a em(T_,[(L(V,x))+]) Pm(s) = e Then, the conditions of Lemma 2.1.2 are fulfilled, since 3 (s) < De'3[[(L(Vx))+I <_ DIIL e"'IIVx=I = DjIL+ Ia(s) Lemma 2.1.2 implies that, for every t E (0, 7), e^wtllVtx < DjIIxjeDIL+I* Since r was arbitrary, D > 1, and Voa = Tox = x, 2.37 is satisfied for every t > 0 and x E B+. Q.E.D. = Ttx + (MP) 1(0,) Tt.[(L(V.x))]ds Tt[(L(V,x))]ds . For the existence and uniqueness of the solution we need Lipschitz condition. We will say that L : B+ B is locally Lipschitz if L(O) = 0, and, for every K > 0, there exists C(K) > 0, such that II Lx Ly II < C(K) x y \ (2.39) whenever x,y G B+, and Ijj1 K, IlyI < K. Notice that, if x(t) : I ) B+ is bounded, i.e., there exists K > 0 such that, for every t E I, jIx(t)l < K, then, by 2.39, L[x(t)] is bounded (by C(K) K) on I. Therefore, if L is locally Lipschitz, then it is enough to keep as our starting assumptions (see 2.28) that L is Mmeasurable. Of course, every locally Lipschitz operator is also continuous. Theorem 2.1.3 (Uniqueness and semigroup property) If L is locally Lipschitz, then there is no more than one solution of 2.28. If ( Vt; t > 0) is a solution of 2.28 and L is locally Lipschitz, then (Vt) is a semigroup. Proof. Assume that there are two solutions, say (Vt) and (Ut), of 2.28. We fix xz B+ and 7 E (0, +oo). Then there exists K > 0, such that max{ IVtx;, IIUtxll} < K, for every t E [0, ]. By 2.28, 2.20 and linearity of T, we obtain, for every t E [0, 7], Vt UX = (MP) J) T,[L(Vt,x) L(Ut,x)] ds = = (MP) ft) Tt_,[L(Vx) L(U,x)] ds It follows that tf (m o Tt_,)[L(Vx) L(Ux)] ds < II V UtX  = sup . mEM\{o} tlim t (m o Tt_.,)[L(Vx) L(Ux)] ds (2.40) Ssup ds (2.40) meM\{0} o Iml Consider now a(t) = jI Vtx Utx I on (0, 7), and : (0, t) [0, +oo), where = (mo Tt_)[L(Vx) L(U,x)] Ilmll By 2.40 we get a(t) < C(K)2K. Dexp(wr), i.e., a is bounded. Since, by Lipschitz condition and 2.25, ,(s)<  ,,ITt.I. II L(Vax) L(Ux) II < < De(') C(K) I V,x Ux I < De" C(K) a(s) Lemma 2.1.2 implies that a(t) = 0, for every t E (0,7). Since Vox = Uox = x, and, x and 7 were arbitrary, we obtain Vt = Ut, for every t > 0. A similar idea, with a little more computation, leads to the semigroup property of (Vt). Consider x E B+, and s,t E [0, +oo). By 2.22 and 2.24, we get, for every SE [0, t], T,(V,x) =, T. (T + (MP) ) T[L(V.x)] dv = = T,T, + (MP) 0,,) T,.+[L(V.,x)]dv = = T,,,x + (MP) fu T[L(V+,_,x)]dw where the last equality follows from the fact that it is true for every m c M. It follows that, for every u E [0, t], Vx VuVX = (MP) 0 Tz[L(Vu+,z_,)]dz (MP) ,,) T,[L(V,,_.x)]dw (MP) f,) T,[L(V.(V.x))]dy = = (MP) J) Tz[L(V(+,,) L( (V.x))]dz = = (MP) J0,) Tu.[L(V.+,x) L(V,(Vx))]dw We are now in a similar position as in the case of uniqueness. We consider functions a : (,t)  [0, +oo), a(u) = I V.+,x V,(V.x) I, and 3, : [0, u] [0, +oo), (m o T.) [L(V.+ax) L(V,(Vzx))]I Ilmll By using K = max{max{ lV,+,xll, IIVw(Va)}; w E [0,t]} and Lemma 2.1.2, we obtain, as we did before, that a(u) < C(K)2KtD exp(wt) and ,8,(w) < Dexp(wt). C(K) a(w). Hence, a 0. In particular Vt+,x = VtVax. Since s,t and a were arbitrary, (Vt) has the semigroup property. Q.E.D. Let us show that the perturbed semigroup (Vt) behaves nicely with respect to further linear perturbation. This fact will have an impact on some of our results later, but the special case of this property is needed to prove the existence of the solution of 2.28. Let ( St; t > 0) be a linear, MPettis integrable semigroup (it means that (St) satisfies all the conditions of Definition 2.1.3 except, maybe, positivity) such that, for every x E B, Stx = Ttx + (MP) t) S,[B(T,_,x)]ds (2.41) where B is Minvariant bounded linear operator. Recall (Lemma 2.1.1) that such an operator is Mmeasurable. Thus L B is Mmeasurable operator on B+, such that (L B)[x(t)] is bounded on bounded subintervals, whenever x(t) E B+ is bounded on bounded subintervals. It shows that the following statement makes sense. Lemma 2.1.3 If ( V; t > 0) is a solution of 2.28, then, for every x E B+ and t > 0, Vt = St + (MP) 0) S,[(L B)(Vt.x)]ds (2.42) Proof. By our assumptions (see 2.26) s , S,[L(Vt_,x)] is MPettis integrable. Thus, we consider the following integrals (MP) (,t) S,[L(Vt.,x)]ds = (MP) Jo) T,[L(V_,x)lds+ +(MP) f() (MP)f(o S[B(T._(L(Vt,x)))]du" ds , where equality follows from 2.41. The condition 2.26 guarantees that (m, u, ) s m{S,[B(T,_,(L(Vt.x)))]} is M (0 B((0, +oo)) ( B((0, +oo))/B(R) measurable, and is bounded on (0, t) x (0, t). Hence, the requirements for Fubini theorem 2.23 are fulfilled. Together with the fact that the first integral above appears in 2.28 we obtain (MP) ft) S.[L(Vtx)]ds = Vtx Ttx+ +(MP) J0,) [(MP) ,t) S,[B(T,,(L(Vt,,x)))]ds] du Notice that S, o B is an Minvariant bounded linear operator. Thus we can apply 2.22 on the integral on the right hand side, to get that it is equal to (MP) J(t)(S. o B) (MP) i) T_,,[L(Vt,x)]ds du = =(MP) Jt)(S o B) [(MP) f(ot) T,[L(V(t,.x)]dw du= = by 2.28 = (MP) t)(S, o B) [Vtx Tx] du = by linearity of S o B = = (MP) ft)(Su o B)(Vtx)du (MP) J) (, o B)(T_,x)du = = by 2.41 = (MP) J (S, o B)(Vtux)du Six + Ttx Finally, it shows that (MP) Jt) S,[L(Vt,x)]ds = Vtx Ttx+ +(MP) ot) S,[B(Vt_,x)]du Stx + T , which, since S, is linear, finishes the proof. Q.E.D. For any a R, an operator B, given by B(s) = ax, is Minvariant, bounded and linear. Also, St = ea tT is a linear MPettis integrable semigroup, which is also positive. Even more, (St) satisfies, for every x E B, (MP) 0,t) e"T,[B(t,x)]ds = (MP) 0t) ae"Tx ds = = aea"ds Ttx = eatTtS Ttx = Stx Ttx If we apply Lemma 2.1.3 on a and a, then we obtain the following statement. Corollary 2.1.1 A semigroup (Vt; t > 0) is a solution of 2.28 if and only if it is a solution of Wtx = e"tTz + (MP) 0,t)(e"' T)[(L + a)(Wt_,x)]ds (2.43) where a E R. In the case of superprocesses, Corollary 2.1.1 has been proved, using integration by parts, in [10], p.58. We will use Corollary 2.1.1 to prove the positivity of the solution of 2.28. To prove the existence of the solution of 2.28, we require several properties on L. We will assign a special name to such L. Definition 2.1.4 An operator L : B+  B is a SOLPoperator (solvable with posi tive solution) if it is an Mmeasurable, locally Lipschitz operator which satisfies either condition a) or condition b) from Theorem 2.1.2, and such that, for every K > 0, there exists a = a(K) > 0, such that Lx + ax E B+ (2.44) whenever x E B+ and III < K. Theorem 2.1.4 If (Tt; t > 0) is a positive, MPettis integrable semigroup, and L is a SOLPoperator, then there is one and only one solution to 2.28. The solution is a semigroup and satisfies boundedness condition 2.37. Proof. It is enough to prove the existence of the solution of 2.28. We will prove it under the assumption b) from Theorem 2.1.2. The case of condition a) is analogous, but simpler than the case b). Without loss of generality we can assume that D = 1. Recall that there is a norm  I' on B, defined by 111' = sup IleWTtll , t>o such that I xaI I 11xi1' < Dlix i, i.e., II I and II 1' are equivalent norms. It follows that IITt I' < exp(wt), and, for every m e M, IIm l' < Ilmll < DImll'. All the requirements in this section are of topological or measuretheoretical nature, except for 2.8 and 2.13. Thus, they will not change when we change to the equivalent norm. Also, we will use 2.8 only in case 0 < x < y. Therefore, to prove that we can restrict ourselves to the case D = 1, we have to prove that II 11' satisfies 2.8 for 0 < x < y, and 2.13. If 0 < x < y then, for every t > 0, 0 < exp(wt)Ttx < exp(wt)Tty, since Tt is positive. By 2.8 for the norm II i we obtain I1 exp(wt)Ttxll Il exp(wt)Tty l, which implies that jIx' < Ilyll'. 2.13 is equivalent to Ixi' = sup{lm(x)I; m E M, Ilmll' < 1} Notice that lxij' > sup{lm(x)i; Ilmll' < 1}, since Im(x)l 5< lmjl' lxl', for every bounded linear functional. Using 2.13 for i II we obtain ix' = sup IlewtTtll = sup sup ew(m o Tt)(x)l t>0 t>0 IlmiS For every t > 0 and m e M, Ilmll < 1, h = exp(wt)(m o T() is in M, since Tt is Minvariant and M is a vector space. Using the equivalence relations for the operator norms i  and II ' we get Ilhll' < e wllmll' ITt' < e" 1 e" = 1 Hence 11x\' < sup{jh(x); h E M, \\h\' < 1} , which proves 2.13 for  '. Therefore, we can restrict our proof to the case D = 1. Fix a strictly positive number 7 and vector x e B+. Let K be equal to iix\ e , where IIL+Il is defined by 2.36. Let a = a(K), defined by 2.44. By Corollary 2.1.1 it is enough to prove the existence of the solution of 2.43. We define inductively Vox 0, and, for every k E N U {0}, S tTtx + (MP) t)e'T,[(L + a) Vt,z]ds (2.45) We claim that, for every k N U {0}, and, for every t E [0, ], Vkx is well defined, 0 < Vkx, 1Vtkz < \x\ exp((w + HL+)t) < K, and [0,7] 9 t  Vtk is Mmeasurable. Obviously, for k = 0 the statement is correct. Assume that the statement is true for k. Since L + a is Mmeasurable and locally Lipschitz, it follows that (m, t, s) t m[e"T,[(L + a)Vtz]] is M 0 B([0, 7]) ( B([O, t])/B(IR) measurable, and MPettis integrable with respect to s. Therefore, the MPettis integral in 2.45 exists and Vtk+lx is welldefined. By Fubini theorem (m, t) m Vkt1 is M i B([0,r])/B(i~) measurable. Since 1IVt8x < K, and Vt,x > 0, it follows, by 2.44, that (L + a)(V,k,x) > 0. By 2.18 and the positivity of T, we obtain Vktx, > 0. Using the positivity of T, one more time we obtain 0 < Vk+x < eTt + (MP) 0,e"aT[(LVa.x)+ + aVtx]ds By 2.8 for positive vectors we obtain iVtk+lx I < eIITtx + II(MP) 0t) easT.LV x), + aV, ds First, we estimate Iea'T.[(LVt. )+ +aVt,xzj to be less or equal to ea ew"(IL+II + a) xI exp[(w + 1IL+ 1)(t s)], and then apply 2.21 and 2.25 with D = 1, to obtain IIVk+'Xll 11:11 (wa)t + e(+IIL+II)t (IIL+ + a)e(IIL+II+a)ds) = = II [e(wa)t + e(w+IIL+ll)t (e(l'L+ +a)I)] = = II lle(W + Ill) By mathematical induction, the statement is true for every k > 0. Let C = C(K) + a, where C(K) is a Lipschitz constant defined by 2.39. By 2.45, we obtain VII Vkl tk I = = (MP) f(t)e"aT,[(L + a)Vtx (L + a)V tx]ds The norm ee"T,[(L + a)V~.x (L + a)Vt.x] < e"' C Vtk Vxt I , since all the vectors involved are bounded by K, and L is locally Lipschitz. Let K, be equal to exp(w air). Applying 2.21 recursively k times we obtain II Vk+x Vtkx I < K Kk Ck. S ds / d.s2 dsk = K KkCk Jo Jo 0 k! Therefore { Vtk; k > 0 } is a Cauchy sequence in B+ C B. B+ is a closed subset of a Banach space B, thus there exists a limit (in the norm sense) Vtx = lim Vkx B+ k00oo Since IVtkzll < K, and m[Vtx] = limkjom[Vtkx], it follows that t Vtx is M measurable and bounded on [0, ]. In particular, it shows that s * ea'T,[(L + a)Vt_,x] is MPettis integrable, and, for every m E M, m [(MP) /) e'T,[(L + a)Vt.,x]ds = = Jm [eT,[(L + a)Vt,x] ds By the dominated convergence theorem, the integral on the right hand side is the limit of [ m [eaT,[(L + a)Vkxl] ds , for every m e M. Since {Vtk+1l etTtx} converges strongly to Vtx etTte, and M separates the points in B, it follows that Vtx satisfies 2.28 for exp(at)Tt and L + a. By Corollary 2.1.1 t V Vtx is the solution of 2.28 on [0, T], for x E B+. Since the uniqueness theorem (Theorem 2.1.3) is valid for every x E B+ and every 0 < 7 < +00, we can define Vtx for every x E B+ and every t > 0, so that Vtx satisfies 2.28, Vtx E B+, t * Vtx is Mmeasurable and bounded on bounded subintervals. Q.E.D. Notice that if the condition a) from Theorem 2.1.2 is satisfied, and condition 2.44 is not necessarily satisfied, the same proof as above will show the existence of the solution of 2.28, even when (Tt) is not positive. However, in such a case we can not obtain the positive solution, in general. Let (Tt; t > 0) be an MPettis integrable semigroup, i.e., (Tt) satisfies conditions in Definition 2.1.3, except positivity. Let B : B B be an Minvariant bounded linear operator. Then B restricted on B+ is locally Lipschitz (actually Lipschitz), Mmeasurable, and satisfies the condition a) from Theorem 2.1.2. It implies, as we just said in the paragraph above, that there is a solution (St) of 2.28, which is the limit of approximations (Sf), given by 2.45, for a = 0 (and therefore Corollary 2.1.1 is not needed in this case) and L = B, but (St) is not necessarily positive. Since all the operators involved are linear, then St is linear, and therefore St is linear, too. Thus, it can be extended on B = B+ B+. It shows that there exists a semigroup ( St; t > 0) of linear operators such that, for every x E B, (m, t) m[Stx] is M 0 B(R+)/B(R) measurable, 'lS, l < Dexp((w + IIBI)t), and, Stx = TtX + (MP) o) T,[B(St,x)ds (2.46) Also, Stx = limk_o Sx, where S, = 0, and S+lx = Ttx + (MP) 0,t) T,[B(SLx)]ds (2.47) Let us restrict our attention to the special case, when B is separable and M = B*. As we have seen in Example 2.1.1, in such a case all the functions involved are actually strongly measurable, and, therefore, Bochner integrable. In particular, our semigroup (St), given by 2.46, is a semigroup of bounded linear operators which satisfies 2.25 and t t4 Stx is weakly measurable. As we have shown in Example 2.1.1, it implies that (Se) satisfies all the requirements of Definition 2.1.3, except, maybe, positivity. Therefore, there is a solution (Wt; t > 0) of the equation WtX = Sx + j) S,[(B)(Wetx)]ds (2.48) where the integral involved is the Bochner integral. If we rewrite 2.46, using Bochner integral and linearity of the operators involved, we obtain that Ttx = Stx + ,t) T.[(B)(St,.x)]ds These are exactly the conditions of Lemma 2.1.3. Since (B) (B) = 0, we get, for every x E B+, WtX = Ttx, and, because of linearity, it follows Wt = Tt. It shows that S.= To + S,[B(T,_,w)]ds , Stx = Ttx + ot) S.[B(Tt)x)]ds which is exactly 2.41. Hence, we just proved the following statement: Corollary 2.1.2 If B is separable and (Tt; t > 0) a semigroup of bounded linear operators on B, which satisfies 2.25, and t i4 Ttx is weakly measurable, then, for every bounded linear operator B : B  B, there exists a unique semigroup ( St; t > 0) of bounded linear operators on B, such that t '+ Stx is weakly measurable, and, for every t > 0, IIStll < D e(W+llI)t (2.49) and (St) satisfies, for every x E B, Stx = Tx + o S,[B(Tt_x)]ds= = Tzt + o T[B(St,x)]ds (2.50) where all the integrals involved are Bochner integrals. Moreover, if (Tt) is also posi tive, and B satisfies 2.44, then ( St; t > 0) is positive. Let us show that some of the measurability conditions given in this section are much simpler in the separable case, then in general case. Recall that in the separable case the calgebra of Borel sets with respect to the norm topology is equal to the smallest aalgebra on B such that all mappings.zx f(x), f E B*, are measurable. Consider a mapping (z, f) . f(x) on B x B*. It is continuous with respect to the first variable, and M measurable with respect to the second variable. Since B is separable metric, the mapping (x, f) f(x) is B(B) 0 M /B(R) measurable. This fact enables us to prove the following proposition. Proposition 2.1.1 Let B be a separable Banach space and M = B*. Then a vector valued function t x(t) is Mmeasurable if and only if it is weakly measurable. If L : D(L) C B  B is a continuous operator then it is Mmeasurable. In particular, every locally Lipschitz operator is Mmeasurable, and, if B is a Banach lattice, z x+ is Mmeasurable. Proof. If t z(t) is weakly measurable then it is B(I)/B(B) measurable. Then (t, f) (x(t), f) is B(I) M/B(B) 0 M measurable. Finally, (t, f) (x(t), f) f(x(t)) is a composition of measurable functions, i.e., it is B(I)(M /B1(R) measurable. The reverse statement is always true. If t 4 z(t) E V(L) is Mmeasurable, then it is weakly measurable, i.e., it is B(I)/D(L) n B(B) measurable. Since L is continuous, it is D(L) n B(B)/B(B) measurable, which shows that t 2 L(z(t)) is weakly measurable, and, therefore, Mmeasurable. The last statement follows since every locally Lipschitz operator is continuous, and z z+ is continuous in Banach lattice. Q.E.D. We will finish this section with a remark on condition 2.44. As we have seen it is a sufficient condition for the positivity of a perturbed semigroup. Notice that for some classes of operators this is also a necessary condition. More precisely, let B = R', where a = (zx,...,an) < y = (yl,...,y n) if x, < y( for i = 1,...,n. Let A : R"  R" be a linear operator, i.e., A is a matrix (a j),J=1. Then there exists a uniformly continuous semigroup A (tA)k S = etA = k! k=O which satisfies the equality Stx = x + A(St,x)ds We claim that the following three statements are equivalent: i) St is positive, for every t > 0 ii) A satisfies 2.44 iii) for every i,j E {1,...,n}, i f j, ai, > 0 By Theorem 2.1.4, ii) implies i). iii) implies ii), since a = maxi{la il} will give Ax + ax > 0, for every x > 0. Let us prove that i) implies iii). Suppose contrary to the claim that there exist io and jo in {1,..., n} such that io # jo and aojo < 0. We consider x = (x1,..., x,) such that xi = 0, if i Z jo, and a0 = 1/(aj00). Then zio = 0 and (Ax),o = 1. We define f(t) = (Stx)i,. Then f(t) is a differentiable function, f(0) = xi, = 0, f'(0) = (Ax)i0 = 1, and f'(t) is continuous. Hence, there is e > 0 such that on [0,e] f'(t) < 0, and, since f(0) = 0, we must have f(t) < 0 on [0, e]. This contradicts i). Hence, the proof is complete and all three statements are equivalent. 2.2 Supersemigroups and Superprocesses In this section we will define the main notion of this work. We will follow the approach developed by P. Fitzsimmons and E.B.. Dynkin, and show that their results are special cases of results from section 2.1. Let (B, jj , <) be a Banach space and a linear lattice with normclosed positive cone. Let M C B* satisfies 2.102.13, and consider a measurable space (M+, M+), where M+ = M n B*'+, and M+ = M+ n, M. Following Dynkin [18] we will say, for every measure Q on (M+, M+), that x e B+ em(x)Q(dm) (2.51) JM+ is the Laplace transform of the measure Q. Let (Tt; t > 0) be a positive, MPettis integrable semigroup, as defined in Def inition 2.1.3, and let L : B+ B be a SOLPoperator (see Definition 2.1.4). Then there is a unique solution (Vt; t > 0) of 2.28, which is a positive, (in general nonlinear) semigroup (see Theorem 2.1.4). Definition 2.2.1 A family of kernels Qt(m,dn) on the measurable space (M+,M+) is a (Tt, L)supersemigroup if, for every m E M+ and x E B+, /M e()Qt(m,dn) = em(v' (2.52) The existence and the uniqueness of such semigroups in the generality of Definition 2.2.1 is not known. However, in some special cases it has been studied by several au thors (see [10] for detailed bibliography). The case of (Tt, L)supersemigroup, where (Tt) is a Markov semigroup, and L is, so called, branching mechanism has been stud ied by Watanabe, Dawson, Dynkin, Fitzsimmons and others. We will restrict our consideration to this situation. Let (E, E) be a standard measurable space, i.e., there is a Polish space (Z, B(Z)), with Borel aalgebra, which is Borel isomorphic to (E, ). Recall ([6], Theorem 8.3.6.) that any two Polish spaces of the same cardinality are Borel isomorphic, and that a Polish space is either countable or has the cardinality of the continuum. It follows that a measurable space is standard if and only if it is Borel isomorphic to a compact metric space with Borel aalgebra. Therefore, we can introduce a metric on E, such that E becomes compact and is the aalgebra of Borel sets with respect to the metric. Notice that our notion of a standard space is equivalent to the notion of Lusinian measurable space in [54] (our terminology is from [6] or [18]). We will denote the set of (respectively, bounded, bounded nonnegative) real valued Emeasurable functions on E by (respectively, bW, bpC) E. Then B = bW with the supremum norm, and with the partial order inherited from the real line is a Banach lattice, where B+ = bp. If M is the space of all finite signed measures on (E, E), then M satisfies 2.102.13, and M+ is the set of all finite (positive) measures on (E, E). See Example 2.1.2 for all these results. For f E b and p E M we will denote the integral fE fd1i by (i, f). Remark 2.2.1 There exists a metric d on M+ (so called Prohorov metric) such that d(i,, It) + 0 if and only if i,, )L; where t,, ' pt means weak (Bernoulli) convergence, i.e., for every bounded and continuous f, (fn, f) (ip, f). It is important that we understand statements about (M+, M+) properly. If we talk about measurability properties of M+ then no topology (or metric) is needed, since the definition of M+ is purely measuretheoretic. But, if we talk about the Prohorov metric (or weak convergence) then we assume certain topology on E. If we do not specify this topology, it means that we consider E as a compact metric space described above. Notice that in this case M+ with Prohorov metric is a locally compact, complete, separable metric space and its Borel aalgebra is exactly M+. In particular, it shows that (M+, M+) is a standard measurable space. For the proofs of results mentioned above see, for example, [7], pp.607638. However, sometimes we can consider E with its "natural" topology (if it exists, of course), which is not necessarily compact. In such a case, we have to deal with topological properties of M+ separately. Let = = (t; t > 0) be a timehomogeneous Markov process, with state space (E, ), transition function pt(x, dy), and corresponding semigroup (Tt; t > 0). We assume that, for every A E (t, z) i pt(x, A) is measurable. (Tt) is a contrac tion semigroup, and, as we have seen in Example 2.1.2, (Tt) is a positive, MPettis integrable semigroup. Let b E b, c E bpE, and n(x, du) be a positive kernel from (E, E) to the measur able space of positive real numbers ((0, +oo), B((O, +oo))) such that x  un(x, du) bp (2.53) In particular, 2.53 shows that there is a constant k > 0 such that, for every x E E and e > 0, k n(x, [e, +oo)) k (2.54) We define a branching mechanism as a function T : E x R+ R of the form (x, A) = b(x)A c(x)A2 + 0(1 eA)n(x, du) (2.55) For f E bpg we define If E by (I f)(x) = (x, f (x)) (2.56) Using 2.53 and the fact that, for z > 0, 1 ez < z, we obtain that, for every f E bpE, 1f11 11 bl + c. 11ft11i + un(x, du)\) (2.57) Notice that the factor cl Ilf prevents us from concluding that k is bounded, but 2.57 does show that T : bp  bE, and T0 = 0. Notice also that directly from the definition 2.55 follows that T is Mmeasurable. Let us prove that I' is locally Lipschitz. Applying the mean value theorem on z exp(z), we obtain I (Tf)(x) (g)( ) < I f(x) g(x) I (Ib(x)) + c() f(x)+g(x) i+ 0 un(x, du)) which, together with 2.53, shows that 'I is locally Lipschitz. It is possible to rewrite a branching mechanism in the form /+00 (z, A) = b(x)A c(z)A2 + (1 eX" Au)n(x, du) (2.58) where /+00 b(x) = b(x) j un(x,du) (2.59) Notice that the assumption 2.53 was crucial here, and that it shows that b E bE. Since 1 e^" Au < 0, for A > 0, it follows that, for every f E bp&, (f) = (b) f (2.60) This, together with our choice of Banach lattice bW, shows that an operator 'I satisfies condition b) from Theorem 2.1.2, and that I+II = l l. Let K > 0 be a positive number. The fact that the derivative d (x, A) < Ib + 2AIcI + i +0 un(x, du) implies that a =a(K) sup {(,) A) xEE,AE[0,K] (dA is finite and nonnegative. Then, for every x e E, A H* (zx, A) + aA is nonnegative and increasing on [0, K], which implies that T satisfies condition 2.44. Hence, we proved that k is a SOLPoperator. It follows now, from Theorem 2.1.4, that there exists one and only one solution Vt : bpE bpE of an integral equation t (Vf)(x) = (Tf)(x) + T[TV_,.fI(x) ds (2.61) and that (Vt; t > 0) is a semigroup, (t, z) (Vtf)(z) is measurable, and, for every f e bpE, (2.62) IV1fll < eilll"lflI . This result was obtained by Fitzsimmons, [28], using similar method as our it eration method in section 2.1., but applied only to the case of bp and %k, and by Dynkin, [18], in the case of $(x, A) = A2, and by different method. Here we just proved that their results are special cases of our general result given in section 2.1. However, the semigroups of this type were studied in connection with measurevalued processes for the first time in Watanabe, [56], and in several works by Dawson (see [10] for complete bibliography). It must be mentioned that the basic ideas for such development were already given in Jifina, [36]. Following Dynkin's terminology we will give a special name to a (Tt, ')supersemigroup. Definition 2.2.2 A (Tt, %)supersemigroup, where %I is a branching mechanism, and (Tt) is a semigroup which corresponds to a Markov process (&t) is called a Dawson Watanabe supersemigroup. Markov process (Xt; t > 0) with the state space (M+,.'M+) is called a Dawson Watanabe superprocess over (It), with branching mechanism T, if its semigroup is a Dawson Watanabe supersemigroup. We will use the same name (DawsonWatanabe supersemigroup) to indicate the transition function Qt(,, dv) on (M+, M+), and the actual semigroup of operators (QtF)(p) IM F(v)Qt(A, dv) (2.63) Also, when it is clear what 4 and ((t) are, we will refer to (Xt) as to a superprocess only. The question of the existence and uniqueness of the superprocess comes immedi ately into consideration. Notice that once we have a supersemigroup Qt(Gi, dv), then, since (M+, M+) is a standard space, Kolmogorov existence theorem guarantees the existence of the superprocess (Xt). Of course, for a moment we do not say any thing about regularity properties of (Xt). We will come to this in the next section. Therefore, the question is the existence and uniqueness of Qt(tI, dv), given by 2.52. Notice that the Markov property of Qt(~p, dv) is the consequence of the semigroup property of (Vt). Notice also that /L i exp((jz,Vtf)) is M+measurable. An ap plication of the monotone class theorem for the multiplicative systems of functions shows that i Qt(p, dv) is M+measurable i.e., Qt(u, dv) must be a kernel, if it exists. VtO = 0 shows that every Qt(p, dv) is a probability measure. That the Laplace transform determines a measure uniquely (on a standard space) has been proved in [18], pp.264265. Therefore, it is enough to prove that for any p and t > 0 there exists a measure Qt(I, dv) which satisfies 2.52. The solution to this problem has been given in [28]. However, some parts of the construction from [28] are more demanding than it may look at first sight. This and the wish for completeness of the work led us to include these proofs here. Notice that (B+, +) forms a twodivisible semigroup, i.e., for every x E B+, there exists y E B+, such that x = y + y (of course, y = (1/2)x). Hence, the Laplace transform is a positive definite function on the semigroup B+ (see 2.51). The theory of positive definite functions on a twodivisible semigroup (S, +) is developed in [2]. It has been proved there that for every bounded positive definite function p : S R there exists a unique positive Radon measure 7 on S, such that s(s) = p(s)7(de) (2.64) where S is the set of bounded semicharacters, i.e., of the functions e : S * [0,1] such that e(0) = 1, and e(si + s2) = e(si) e(s2). Moreover, if p(0) = 1, then 7 is a probability measure. Recall also ([2], Theorem 3.2.2.) that positive definite functions are negative exponentials of negative definite functions, i.e., of the functions S: S  R, such that E aiajO(s, + s,) < 0 (2.65) t,=l whenever {ai,...,a,,} C R sum to zero, {sl,... ,,Sn} C S, and n > 2. Therefore, in application of these theories on the solution (Vt) of 2.28, one has to prove first that x m[Vtx] is negative definite. In terms of branching mechanism and in the notations of bp, one has to prove that f (Vtf)(x) is negative definite, for every x e E (do not confuse x E E and x E B !). Remark 2.2.2 The statement about negative definitness of f F+ (Vtf)(x) is given in [28], p.341, and it is suggested there that the proof is standard. Through private communication with the author of [28] we pointed out some possible difficulties in the proof. After some discussion both sides offered correct proof (which happened to be the same on both sides). This is still a standard proof, but different from the first proof that was suggested. We present the proof below.o Proposition 2.2.1 Let B+ = bp, and L = T. If (Vt; t > 0) is a solution of 2.28, then, for every t > 0 and x E E, f * (Vtf)(z) (2.66) is negative definite on (bpE, +). Proof. Consider, for every m E N, the branching mechanism '.(, A) = b(x)A + (1 e"A )ii(x, du) (2.67) where f(x, du) = n(x, du) + 2m2c(x)6)1_ (2.68) where Sa} is the pointmass at a. Notice that, for every x and A, Im(x, A) Q I(z, A), when m t +oo. Hence, for every f and x, there is a subsequence {mk} such that (Wtkf)(x) (Vtf)(x), where (Wm; t > 0) is the solution of 2.28 for L = mQ. Therefore, it is enough to prove negative definiteness of the solution of 2.28 where L is the branching mechanism of the form 2.67. Notice that, because of 2.53, such branching mechanism can be written in the form (x, A) b(x)A + J (1 eA")9(x, du) (2.69) For a fixed x E E, the function above is a Bernstein function (see [3]). The com position of a Bernstein function with a negative definite function is again negative definite (see [3], and [2] p.114). If ( Wt; t > 0) is the solution of 2.28 with the branch ing mechanism of the form 2.69, then (Wtf)(x) = lirrm oo(Wtf)(x), where W' are defined inductively by 2.45. Since Wo 0 is negative definite, and linear function (Ttf)(x) is negative definite, we prove inductively that (WT'f)(x) is negative definite, for every I (applying the above fact about Bernstein function). Q.E.D. Remark 2.2.3 There is a possibility for an easy mistake in the proof above. In some cases a function, say 0, can be regarded as a function on a group and on a semigroup, as well. Typical example is the function 0(x) = x2. 0 is the negative definite function in the sense of a semigroup (R+,+), but 0 is not the negative definite function in the sense of a group (R, +). However, when we compose 0 the situation changes. 0 composed with the negative definite function in the sense of a semigroup (R+, +) is not necessarily a negative definite function on (R+, +). For example, x F4 ln(1 + x) is a negative definite function on (R+, +) (to prove that consider the Laplace transform of the rdistribution), but x * [ln(1 + x)]2 is not.O The last part of the construction of Qt(l, dv) is given in [28]. For the sake of completeness we will present it here (in some nonessential details our argument will be different). Notice that, for every x E E, limn_.o(Vt;)(x) = 0, by 2.62. By Proposition 2.2.1 and 2.64 there is a probability measure 7 on S (where S = bpE), such that = P(f)y() (2.70) and is unique. Since p(l/n) = e(1)1/", it follows that 7 is concentrated on S+ = {e gS (s) > 0, for every s S } If f, \ 0, f, E bpE, then (Vtf,)(x) \ g(t) > 0, since Vt is monotone, by 2.70 (in the sense that if f,g E bpE, f < g, then Vtf < Vtg). Using 2.28 we get g(t)= T,['g(t s)]ds By the uniqueness of the solution of 2.28, we obtain g(t) 0, i.e., Vtf, \ 0. By 2.70, e(fn) / 1 7(a.e.). If we fix f E bp, g E S+, and, for every t > 0, we denote e(tf) by g(t), then g(0) = 1, g : [0, +oo)  (0,1], and g(t + s) = g(t)g(s). Hence, g(t) = exp(at), where a = In o(f) > 0. Now, let us define T: bpE  bB(S+) by (Tf)()= Ine (f) (2.71) where B(S+) is the oalgebra of Borel sets on S,. We just proved that T is additive, homogeneous for positive real numbers, and, if f, \ 0, then Tf, \ 0 7(a.e.). Very simple adjustment of the theorem in [31] for the case of standard spaces, shows that there exists a kernel K on S+ x bounded and positive, such that, for every f e bp, Kf = Tf 7(a.e.). (2.72) Hence, for every g E S+ there is a measure K, in M+, and, by definition of M+, e K, is B(S+)/M+measurable. Let Q be the image measure of under Lp Kp. Then, by 2.70 and 2.72, e(U>f) = + e(vf)Q(dv) (2.73) and the construction of a Markov transition function Qt(i, dv), which is also a su persemigroup, is complete. In other words we proved the following theorem: Theorem 2.2.1 (Dynkin, Fitzsimmons) Let (E, E) be a standard space. For every timehomogeneous Markov process ( on E, such that (t, x) pt(z, dy) is measurable, and, for every branching mechanism 9, which satisfies 2.53, there exists one and only one supersemigroup Qt(I, dv). Remark 2.2.4 Locally Lipschitz property was crucial in the construction of the su perprocesses. We were using 2.53 to obtain it. However, it is worth mentioning that locally Lipschitz property of T follows under weaker condition than 2.53, i.e., under the condition u n(x, du) + un(x, du) e bpS 0O 1 But in this case we can not transform 2.55 into 2.58 and vice versa. Therefore, it leads to different constructions where our general approach, developed in section 2.1., cannot be applied so easily. For the construction of superprocesses under this weaker condition we refer to [10], Chapter 3. Let us also mention that in [28] the author assumes 2.53 and one more condition, i.e., that x u2n(x, du) E bpE , but he uses it only for the regularity properties of the superprocesses. For the con struction of the supersemigroup it is not needed.0 Let us mention that there is a nice (and important, as we will see later), proba bilistic interpretation of the branching mechanism T. Notice that, for every x E E, A W qI(z, A) is a LogLaplace function of an infinitely divisible distribution (see [27], XIII.7.). More precisely, e('') = E (e(D.+N.+P)) (2.74) where D,, N,, P, are independent random variables with values in R, such that D, is a pointmass at b(z), N, is a centered normal distribution with Var(N,) = 2c(x), and P, is a positive infinitely divisible distribution with finite expectation E(P.) = un(x,du) (2.75) Recall also (see [35] for example), that there is a unique temporally homogeneous Levy process, i.e., a process with stationary independent increments with no point of fixed discontinuity and whose sample paths are cadlag, say (Lk(")), such that e = E e L')) .= (2.76) If T(z,A) = A then (Lt (")) is a uniform motion to the left. If T(x,A) = A2 then (Lt ("')) is a standard Brownian motion. If {(x, A) = 1 e then (L*( '')) is a Poisson process. We will apply Lemma 2.1.3 on superprocesses to obtain an interesting result, which suggests that in some cases we can study always the same (and extremely sim ple) Markov process, and change the perturbing operator accordingly, to get Dawson Watanabe superprocess. Let us denote the trivial semigroup, which consists only of identity operators, by (Idt; t > 0). This semigroup is defined on bE, but also it is defined on any subspace of b. To this semigroup corresponds a trivial Markov process on (E, ), which "stays forever at the point". Let us denote this Markov process by (dt; t > 0). Then, for every t > 0, dt do. Theorem 2.2.2 Let ( be a Markov process, such that there exists an Minvariant, bounded, linear operator A, such that, for every t > 0, Tt = exp(tA). Then there exists a DawsonWatanabe supersemigroup over (, with branching mechanism T, and is equal to a (Idt, E + A)supersemigroup. Proof. By our assumptions, for every G E E, pt(x, G) = P(( E G) = (TlG)(x) = E (A~nl)(X) n=.O " which shows that (t, z) '4 pt(x, dy) is measurable. By Theorem 2.2.1, there ex ists one and only one supersemigroup Qt(L, dv), such that its Laplace transform is exp((A, Vtf)), where, for every f E bp, Vf = Ttf + (MP) (ot) T.[W(Vtf)]ds As we have seen in Example 2.1.2 (Tt) is a positive, MPettis integrable semigroup, and Ttf = Idtf + (MP) Jt) T,[A(Idt_,f)]ds which implies, since A and T, commute, Idtf = Ttf + (MP) j) Id,[(A)(Tt,f)]ds Since A is Minvariant, we are in the situation of Lemma 2.1.3, which implies that Vtf = Idtf + (MP) 0t) Id,(i (A))(V,)]ds = =f + (MP) [0t)(T + A)(Vf)ds (2.77) Q.E.D. An example of a Markov process which satisfies Theorem 2.2.2 is a Poisson process. Another interesting class of processes which satisfy Theorem 2.2.2 consists of Markov chains with differentiable transition probabilities. We will describe this case more carefully here. Let E = {1,2,...,n} and E = 2E. In this case B = b is equal to R", and B+ = bpE is equal to R+ ={(x,...,xn) E R" Ix > 0, for every} However M = B* is also equal to Rn, and M+ = R+. Notice that M = B(R") and M+ = B(nR"). To distinguish between measures and functions we will use notation f, g, .., f = (fi, fn) for functions, and ,., = (L1,.., in) for measures. Of course I = I({i}), and f, = f(i), and (i, f) = =1C ifi. In this case Tt is a matrix with elements pi(t), i,j E E, and the measurability condition, on (t, x) pt(z, dy), is equivalent to the condition that, for every i,j, t 4 pij(t) is measurable. By Theorem 2.2.1, the DawsonWatanabe supersemigroup exists already in this case, but we would like to consider the cases in which requirements for Theorem 2.2.2 are fulfilled. Since we are in a finite dimensional case, these requirements are equivalent to the condition that (Tt) is a Feller semigroup, which is equivalent to the condition that, for every i,j, limp,,(t) = 6j. (2.78) In the following we will restrict our attention to such semigroups (Tt). In such a case there exists a matrix A, an infinitesimal generator, such that A = (aj)n and Tt = exp(tA), and a, > 0, for i Z j, Eai = (2.79) j=1 Compare this condition to the end of section 2.1. A good reference for these results is [5]. In [5] a transition matrix (Tt), which satisfies our conditions, is called a standard transition matrix, and the matrix A is called a Qmatrix. We will refer to (Ti) as a standard transition matrix, but to A as an infinitesimal generator (we use Q for supersemigroups). Typical example of a Markov chain which has a supersemigroup, but does not have a standard transition matrix (and, therefore, does not satisfy Theorem 2.2.2) is, for n = 2, Tt= 0 ; t> (2.80) 2.3 Kac Semigroup and Linear Case In this section we will describe DawsonWatanabe superprocesses over very gen eral class of Markov processes with linear branching mechanism. In this case Dawson Watanabe supersemigroups are related to a familiar notion of Kac semigroup, and they can be used as upper and lower bounds (in some sense) for the general case of branching mechanism. Let (() be a Borel right Markov process on a Lusin state space (E, ) with transition probability pt(x, dy) (see [54]). In this case E is the aalgebra of Borel sets, ((t) is a.s. right continuous, and (t, x) pt(x, dy) is measurable. We will also assume that, for the corresponding Markov semigroup (Tt), Ttl = 1. Let ck bC be a bounded measurable function (not necessarily nonnegative). We define, for every GE 9, kt(x, G)= EX [IG(t) *e fk()du (2.81) Then, by monotone class arguments, kt(x, dy) is'a positive kernel, such that (t, x)  kt(x, dy) is measurable. This kernel defines a semigroup ( Kt; t > 0) given, for every f bpE, by (K f)(x)= f (y)k(x, dy) = E' [f( () e fo k()d] (2.82) Notice that (Kt) is not necessarily Markovian, since k can be negative, but (Kt) is bounded by exp(Ikl t) and is positive. From Example 2.1.2 we obtain that (Tt) and (Kt) are both positive, MPettis integrable semigroups (in the sense of Definition 2.1.3), where M is the set of all finite signed measures on (E, E). Recall (see, for example, [49], Chapter 3) that, in the case of Brownian motion and k > 0, the semigroup (Kt) satisfies, so called, Kac (or FeynmanKac) formula. In this work we will refer to the semigroup (Kt), defined by 2.82, as a Kac semigroup. Consider now a linear operator B : bE  bC, defined by (Bf)(x) k(x) f(x) (2.83) It is obvious that B is an Minvariant operator and an SOLPoperator (see Definition 2.1.4). We claim that (Kt) and (T() satisfy the conditions of Lemma 2.1.3. It is enough to prove the following lemma: Lemma 2.S.1 For every f E bE and x E E, (Kf)(x) = (Tf)(s) + j K,[B(T.,f)](x)ds (2.84) Proof. Using 2.82 and Markov property, we obtain (Kf)(x) = E' [f(t) e ()du ] = Ex [f(6) (I k(u() e fo )d du)] = (Tf)(x) + E [k.(() f (te) e lo'(d du= = (TfX)(x + ) E [ k(G) f () e ufo ( )/,] du = = (Tf)(x) + E E [k() E' [f(&u)] efo(.)d] du = = (Tf)() + 'K.[k. (T.._f)](x)du Jo Q.E.D. Consider now a branching mechanism %Pl(x, A) = b(x)A, where b E bV. Let us denote the corresponding solution of 2.61 by (Wt; t > 0). Let (Kt; t > 0) be the Kac semigroup, where k = b. By Lemma 2.3.1, we can apply Lemma 2.1.3 on these semigroups. Since T1 (bf) = 0, we get Wt = Kt, for every t > 0, i.e., Kac semigroup is a solution of 2.61. Remark 2.3.1 One can think that we have proved already in Lemma 2.3.1 that (Kt) is a solution for 2.61. But it is not so. Compare 2.61 (i.e., 2.28) with 2.84 (i.e., 2.41), and notice that semigroups appear in different "order" under integral. In many cases it will be the same solution, but one has to prove it. However, the solution of 2.61 for l1(x, A) = b(x)A can be computed directly, and this is a known result (see [28]). We just wanted to emphasize that it is also a consequence of our general method developed in section 2.1.0 Once we know that (Kt) is a solution of 2.61, it is easy to construct the correspond ing superprocess. We will introduce some notation first. For a measure p E M+, we denote by A, a point mass at p. Of course, A, is a probability measure on (M+,M+). As usual, the point mass at x E E we will denote by 6,. So, 6, is a measure on (E, ), i.e., 6, E M+, and it make sense to consider A6,. For t Ec M+ we denote by iKt a measure on (E, E), defined by, (Kt,f) = (. ,Ktf) = fE E [f(t) e ob(')du] (dx) (2.85) Obviously, pKt M+. Formula 2.85 immediately shows that we can describe the linear case, %1l(x, A) = b(x)A, completely, i.e., that we proved the following theorem. Theorem 2.3.1 (linear case) A DawsonWatanabe superprocess, over (t) with branch ing mechanism @1(a, A) = b(x)A, is a deterministic Markov process (Xt), defined by Xt = XoKt, where (Kt) is a Kac semigroup (with k = b). The supersemigroup is given by the formula Qt(, dv) = AKt (dv) .(2.86) Consider now a general form of branching mechanism fi(x, A), given by 2.55. It can be rewritten in the form 2.58. Let XI2(X, A) = b(x)A be negative of the positive part of T. Let (Kt) be the Kac semigroup, where k = i. Let (Vt) be the solution of 2.61 for i, and Qt(p, dv) the corresponding supersemigroup. It has been proved in [28], p.342, that the first moment of Qt(,, dv) is given by f f)Q ,d) = (,E [ 0f( te f0(S)ds]) (2.87) These facts lead to one more application of Lemma 2.1.3, which gives results stronger than 2.62 on bounds of (Vt). Theorem 2.3.2 For every t > 0, x E E, and f E bpE, the following properties are satisfied: 0 < (Vf)(x) < (Ktf)(z) = E" [f( t) eo()d'] (2.88) (Vtf)(x)= 0 = (Ttf)(z) = 0 (Ktf)(x) = (2.89) In particular, if b = 0, then 0 < (Vtf)(x) < (Ttf)(x). Proof. Consider T %I2. It is a branching mechanism again, but with the property that, for every x E E and A > 0, (i I2)(X, A) < 0. By Lemma 2.3.1 (Tt) and (Kt) satisfy the conditions of Lemma 2.1.3, which implies that (Vtf)(x) = (KWtf)(x) IK,[( '2)(V_,uf)](x)ds (2.90) Since K,, (%P %i2), and Vt_, are all positive, it follows that (Vtf)(x) < (Ktf)(x), which proves 2.88. To prove 2.89 we should notice first that (Vtf)(x) < (Kf)(x) < (Ttf)(x) exp(1b 11 t), which proves that (Ttf)(x) = 0 =, (Ktf)(x) = 0 ==, (Vtf)(x) = 0 We have to prove only that ((Vtf)(x) = 0 == (Ttf)(x) = 0). If (V f)(x) = 0, then, by 2.52, (v,f) = 0, Qt(6,, dv) (a.e.) (recall that (6,,Vtf) = (Vtf)(x)). It follows that the first moment of Qt(6,, dv) is zero. By 2.87, we obtain that E' [f(6t)e fo'(,)d] =0 . Since b is bounded, the exponential factor is strictly positive, and f(t) e f~6(')da = 0, P' (a.e.) It follows that f((t) = 0, P_ (a.e.), i.e., 0 = Ex [f(6,)] = (Ttf)(x) Q.E.D. Remark 2.3.2 It is a good place to comment on additional requirements on superpro cesses assumed by some authors. In some of his papers E.B. Dynkin (see for example [18]) require, what seems to be a "natural" probabilistic condition, that, for every BE , Probability{(t E B} = Expectation[Xt(B)] Recall that (Xt) is a measurevalued, so above formula makes sense. By monotone class argument, it is equivalent to (~, Ttf) = M(, f)Qt(p,dv) However, formula 2.87 shows that it is the case if and only if b = 0, but not always. In particular, it is true for T(x, A) = A2, which is the case often studied by Dynkin. Despite these facts we will continue to study the general form of %.01 2.4 Feller Condition and Infinitesimal Generator Regularity properties of superprocesses have been studied by many authors. In the case of a compact metric space E, and a branching mechanism T with constant coefficients, the problem has been treated already by Watanabe [56]. His approach deals with Feller properties of semigroups involved. Since M+ is, in this case, a locally compact space, we can deal with Feller property in a nice way. However, very often we deal with Markov processes that are defined on a locally compact Hausdorff space with a countable base. In such a case, M+ is not necessarily locally compact. Several authors offered ways to resolve this situation (see [9], [34], and [52], for example). Usually authors adjust the conditions according to their special needs. In the following text it is general enough, for us, to treat the case when (t) satisfies Feller property. We believe that the most "natural" way is to extend our process onto the onepoint compactification of the state space E, to develop a superprocess over it which has Feller property, and to show that such superprocess actually "lives" on the set of measures on E. Of course, the Feller property of the superprocess on the onepoint compactification is the result of Watanabe. But, it follows also from our general method, developed in section 2.1., as a special case. Let us mention that the problem of regularity properties has been treated in a very general form in [28]. It is shown there that the superprocess is a right Borel Markov process, whenever (6t) is a right Borel Markov process with Lusin state space (E, 8), and T satisfies condition on the second moment (see Remark 2.2.4). The temporally nonhomogeneous case was treated in [17], and generalized even more in [21]. We do not need so general a Markov process (6t), and, on the other hand, it is useful to have Feller property for superprocesses, that is why we will take the approach suggested above. Let E be a locally compact Hausdorff space with a countable base, and E = B(E) aalgebra of Borel sets. Notice that (E, E) is a standard space whose topology is specified and is not necessarily compact. Let EA = E U {A} be the onepoint compactification of E, and A = B(Ea) aalgebra of Borel sets (we leave E intact if it is already compact). On both E and EA we consider metric topologies. Recall that E = EnSa. As usual, Co(E) is the set of all continuous functions on E which vanish at infinity, and C(Ea) is the set of all continuous functions on Ea. Co(E) can be regarded as a subset of C(Ea), and, for every f E C(Ea), f f(A) E Co(E). Let M be the space of all finite signed measures on (E, E), and MA the space of all finite signed measures on (Ea, Ea). By the Riesz theorem M = Co(E)*, and MA = C(E,)*. We will be interested in M+ and M'. Then M+ is a locally compact Hausdorff space with a countable base, with respect to Bernoulli convergence, and MA = B(MA) (see Remark 2.2.1). However, M+ is not locally compact (it is Polish in the topology of vague convergence). We can regard M+ as a subset of M+ in the sense that M+ = {E M+ I (f{A}) = 0 (2.91) Actually M+ is a Gasubset of M+, and M+ = M+ n MA. We assume throughout this section that (t) satisfies a Feller property, i.e., that its Markov semigroup (Tt) is a Cosemigroup on the Banach space Co(E), and that TtlE  1. Then ( can be regarded as a Hunt process (0, T, .F, t, Ot, P)) with infinite lifetime, and A as a cemetery (see [4] pp.4450). Recall that in this situation we extend (Tt) to C(Ea), by (Tf)( (Tt f/E)(x) x E (t ) f(A) X (2.92) Notice that (Ti) is a Cosemigroup on C(EA). Using Example 2.1.1, we conclude that (Tt) ((Tt)) is a positive, M (Ma)Pettis integrable semigroup. We have to restrict our T, too. We assume that b(x) and c(x) are continuous func tions which have limits at infinity, i.e., they can be extended to continuous functions b(z), 6(z) on Ea. We assume also that there exists a kernel fi(x, du) from (Ea, EA) to ((0, +oo),B((0, +oo))) such that, for every x E E, i(x, du) = n(x, du), and, for every a : (0, +oo)  [0, +oo), a(u) < u, and a continuous, the function R+ x E 3 (A, x) a(Au)ia(x,du) (2.93) is continuous. If we take a(u) = u, we get that 2.53 is a continuous function on EA. If we take a(u) = 1 e", we obtain that, for every f E C(EA)+, ', defined by, (f)(x) = b(x)f(x) (x)[f(x)]2 + (1 ef(")")u(, du) (2.94) is continuous on EA. It shows that T (W) is a SOLPoperator on Co(E)+ (C(EA)+). Notice that in the case when b, c, and n do not depend on x, all these conditions are satisfied. By Theorem 2.1.4 there exist (Vt), Vt : Co(E)+ Co(E)+, and (Vt) a solution of 2.28 for I, and (V), Vt : C(Ea)+ . C(Ea)+ and (V) a solution of 2.28 for 4. By the uniqueness Theorem 2.1.3 and 2.92 both (Vt) and (V) are restrictions of the solution of 2.28 for 4' on bpE, and they are related by formula = (Vt f/E)(x) x E (295 )x) = f(A) + f '(A, (V8,(f)(A))ds x = A In particular, it shows that (V(f)(A) = (VOg)(A), whenever f(A) = g(A), and (Vtf)(A) = 0, whenever f(A) = 0. Also, Example 2.1.1 shows that all the functions involved in 2.28 are Bochner integrable, and, therefore, for every f E C(Ea)+, Vtf = Ttf + (B) (ot) T ,[(,8f)]ds (2.96) Hence, the mapping t + Vtf is strongly continuous, and, by 2.94, t '4 Vtf, for f E Co(E)+, is also strongly continuous. Using Theorem 2.2.1 we know that there exists one and only one supersemigroup Qt(p, dv) which corresponds to the solution of 2.28 on bpA. Since C(Ena) C bpEa, it is true that, for every f E C(E)+, M+ e()Qt(I, dv) = e(') .(2.97) Notice that in 2.97 it was important that M' is equal to M+ which corresponds to bpn, as it was defined in section 2.2. Consider the functions Ff : M+ R, f E C(Ea)+, defined by, Ff(J) = e(^'f (2.98) and the family of functions Exp(M+), defined as the linear span of the family { Ff I f C(Ea)+, f strictly positive } (2.99) Since f is strictly positive, Ff E Co(M+) (which makes sense, since M+ is a locally compact space). Since Ff F, = Ff+, Exp(M+) is a nonempty algebra, which sep arates points in M+, and Exp(M+) C Co(M+). By the StoneWeierstraB3 theorem, the closure of Exp(M+) is Co(M+). Recall that, by Theorem 2.3.2 b), Ttf strictly positive implies that Vtf must be strictly positive. If f G C(EA)+ is strictly positive, then, since Ea is compact, there exists a constant a > 0, such that, f > a. Tt is monotone, implies that Ttf > Tta a. Thus, Ttf is strictly positive, and, therefore, Vtf is strictly positive, too. In particular, if Ff E Exp(M+), then F1,! e Exp(MA). Using 2.97 we obtain Qt(Ff) = Ff.f E Exp(M+) (2.100) Since Qt is a bounded linear operator, and Exp(M+) is dense in Co(M+), we conclude that Qt(Co(M )) Co(M ) (2.101) Recall that Vf  f strongly, when t  0+. It follows that, for every p. MA, lim Qt(Ff)( l) = lim (FV,)(p) = F () (2.102) t O+ t0+ Since (Qt) is a contraction semigroup, (QtF)(/I)  F(), for every F E Co(M+). In other words, we have proved that (Qt) satisfies the Feller property. In the case of E compact, and b, c, n constants this result has been proved in Watanabe [56]. Sev eral authors proved the same result later under various assumptions, but essentially Watanabe's method inspired most of these proofs. In this work, we just proved that Watanabe's result is a consequence of our general perturbation theory from section 2.1. and Theorem 2.2.1. However, in section 2.1.we were also inspired by Watanabe's method. Our version of Watanabe's theorem says the following: Theorem 2.4.1 If a Markov process ( with a locally compact Hausdorff state space E with a countable base satisfies Feller condition, and a branching mechanism 41 satisfies 2.93 and 2.94, then (ft, j)supersemigroup (Qt) satisfies Feller condition on a locally compact Hausdorff space M+, which has a countable base. Notice that this result guarantees the existence of a superprocess (Xt) with nice regularity properties. By the basic construction from the theory of Markov processes (see [4], pp.4450), there exists a Hunt process (Xt) with state space MA, which is a superprocess over (t) on EA with branching mechanism '. (Xt) has infinite lifetime, since Qtl,+ = 1 In some sense (Xe) is a superprocess over ({) and %F. But, it has values in MA. We would like to restrict it to M+ somehow. Notice that there is a (Tt, T) supersemigroup Qt(j, dv) on M+, when we consider (Tt) and T on bpE. By 2.91 pL E M+ can be regarded as Az E M+, with Cj({A}) = 0. Thus, for every 1L E M+ and f E Co(E)+, IM e("f)Qt(p, dv) = e('f) = by 2.95 = = e(',v')= e(v')Qt(p, dv) Since the Laplace transform uniquely determines measure Qt(P, dv), it follows that, for every i E M+ and t > 0, Qt(p,M+) = 1, and Qt(pl,dv)/M+ = Qt,(, dv) (2.103) In particular, it shows that, for every t E M+ and t > 0, P" [Xt M+]= 1 (2.104) Although (Xt) is right continuous, we cannot conclude immediately that the super process (Xt) "lives in M+, if it starts there", since M+ is not closed in M'. However, the statement is true, as we will show in the following theorem. Theorem 2.4.2 For every yL E M+, P" Xt E M+, for every t > 0 = 1 (2.105) Proof. By our assumptions 2.93 and 2.94, b is a bounded function, thus, there exists / = [Ibll > 0. Let f = la be a characteristic function of {A} C EA. Then F, defined by F(() = (t, f) = ((Az) is in Mt. Using 2.87 we compute (e'tt)(F)(p) = e *(t,EX [f/( f)e f(()d.]) < < (i,,E" [f(6t)]) = (/, TtlA) = ( L, 1A) = F() . By the dominated convergence theorem lim E" [f( )e fo' = = E" [f(o)] = (TtlA)(x) = l (x) Notice that f(t) = 1(t=A), {A} is closed, and A is a cemetery. It shows that lim f((t) = f(0o) (a.e.), t0O+ which implies the statement about the limit above. By the dominated convergence theorem lim (ePtQ)(F)(j) = F(p) t*O+ Hence, F is /excessive (relative to (Xe)). It follows, see [4], Theorem 2.12 b), that almost surely the mapping t F(Xt)= (Xt, 1) = Xt({A}) (2.106) is right continuous. By 2.104, we get X,({A}) = 0 a.e., for every q Q+. Using 2.106 we obtain that almost surely the mapping t Xt({A}) is identically zero, which proves the theorem. Q.E.D. The fact that (Tt) satisfies Feller condition, i.e., that (Tt) is a Cosemigroup on Co(E), and (Ti) is a Cosemigroup on C(EZ), introduces another powerful "tool" into consideration. It is the infinitesimal generator. Let A be the infinitesimal generator of (Tt), defined on its domain D(A) C Co(E), and A the infinitesimal generator of (Tt), defined on its domain D(A) C C(EA). Recall that f E D(A) if and only if f f(A) E D(A), and (Af)(x) ( A(f f(A))/E(x) x E E (Af)() A(2.107) Denote the infinitesimal generator of (Qt) by 9. Consider the family of functions Exp(M+, D(A)), defined as the linear span of the family { Ff f E C(Ea)+ n 9(A), f strictly positive }. (2.108) In [56] Theorem 2.4 and Lemma 2.2, Watanabe proved that (in the case of constant coefficients b, c, n) Exp(M', D(A)) is a core for 9, and he also computed G(Ff). Since exactly the same proof works in our case without any difficulties, there is no need to repeat the proof here. We conclude that the following statements are true. Exp(MA, D(A)) is a core for (2.109) For every f e C(Ea)+ n 7D(A), f strictly positive, and for every t > 0, Vtf e C(Ea)+ n 7D(A), Vtf is strictly positive. (2.110) For every f E C(Ez)+ n D(A), f strictly positive, g(F )(A) = e,,). (A, Af + f) = =(Ff)() (p, Af bf f+ (1 ef")fi(, du)) (2.111) For the martingalee problem"type characterization of the infinitesimal generator see [28] pp.338 and 354. It is now an easy Corollary of these results, to give a complete characterization of infinitesimal generators of superprocesses over finite Markov chains with standard transition matrix. Let ((t) be a Markov chain with state space { 1,2,... ,n} and a standard transition matrix (Tt) with infinitesimal generator A, described by 2.79. Notice that our state space is already compact, so there is no need to use onepoint compactification. Our branching mechanism is determined by b, c, and n, where now b = (bl,...,bn) E R", c = (cl,...,c,) E R+, and n = (nm(du),...,nn(du)) where each measure ni(du) satisfies condition Suni(du) < +oo (2.112) There are no other restrictions on %. In this case A = A and 2D(A) = R". Notice that, for every f E R+, and a function Ff :R+ n R+, (Fi)(yx) (,, Af) = e CZSift fj ajk) = j=1 \k=1 j=1 k=1 j= J \ ,=1k 8 = (p, A VF) . Similarly we obtain aF (Ff)() (, bf) jb = (, be VFf) j=1 "91Lj and (F)(,Z) Ic/2)= nC," 2 F j=1 "lJ The last term in 2.111 can be rewritten as, (F)(). (, + (1 ef")n(i, du) = = E 1 (F1(A + ue,) Ff(A)) ni(du) where e = ( 0,...,0, 1, 0,...,0). Using Theorem 2.2.2, results about Kac semi group, and results in this section (recall that exponential functions form a core for G; and G is determined on its core), we can summarize properties of superprocesses over Markov chains in the following theorem. Theorem 2.4.3 A superprocess (Xt) over Markov chain (t) with branching mecha nism VT is a Markov process which satisfies Feller condition, and is actually a Hunt process with infinite lifetime whose state space is R The Laplace transform of a supersemigroup (Qt(/i, dv)) is given by R e E=1F IQ(('i,... ,), (dvi,... d.. )) = e s(Vl)., (2.113) where (Vtf) = p,i(t)fj+ j=1 + j P(t s) b(Vf)j c(VCf) + (1 e( "),)n,(du) ds = ot " = f, + ds E pj(t s) (2.114) 0 j=1 +o00 n [b(Vf), c(Vf) + (1 e(Vf)i")n,(du) + E aj,(V f)k i 10 ck=1 The infinitesimal generator g of (Xt) is given by (GF)(i) = (2.115) [ r 2F 9F +0 = b3 + (A VF) + [F(p + uj) F(p)]nj(du) . In the special case, when T = bA, (Vt) is the Kac semigroup and the infinitesimal generator is (GF)(I) = (l, A VF b VF) (2.116) Let us show, in the example below, that even on the level of Markov chains, the superprocess (Xt) over (t) does not necessarily satisfy Feller condition, if (&t) does not satisfy Feller condition. Example 2.4.1 We consider n = 2, and (Tt) given by 2.80, i.e., Tt=[ 0 1] 0 1 Let ('9f); = bfi, i = 1,2; where b E R. Using results about Kac semigroup it is easy to check that (Vtf)i = eb(Ttf)i = ebtf (2.117) and Qt(((A, 1,2); (dv1, dV2)) = A(o,(Al+,2)eb)(ddv, dv2) (2.118) which gives a superprocess Xt = (X'X,X2) to be (Xt, X2) = (0, (XJ + Xo)ebt) (2.119) and it is obvious that such a Markov process does not satisfy Feller condition.0 2.5 Branching Property In this section we will present some easy consequences of the fact that super processes satisfy a branching property. Branching property in this form, i.e., in the continuous state space has been introduced by M. Jifina [36]. Notice that the additive structure of M+ enables us to introduce a convolution operation for measures on (M+, M+). If Q1 and Q2 are probability measures on (M+, M+), then Q x Q2 is a probability measure on (M+ x M+, M+ 09 M+). Since (/, v) pt + v is M+ M +/IM+ measurable, there is a unique measure, say Q1 Q2 on (M+, M+), defined by (Ql Q2)(H) = M+ lx+ H( + V)(Qi x Q2)(d,dv) (2.120) where H E M+. As usual, we will call the measure Qi Q2 the convolution of measures Q1 and Q2. Notice that it is easy to check that the Laplace transform of Q1 Q2 is the product of Laplace transforms of Q1 and Q2. Also, if X1 and X2 are independent M+valued random variables with distributions Qi and Q2, then X1 + X2 has a distribution Q1 Q2. Using the fact that (I1I + Y2, Vtf) = (l, Vtf) + (Z2, Vtf) we conclude that a DawsonWatanabe supersemigroup (Qt(p/, dv)) satisfies the property Qt(GI + 2, dv) = Qt(A, dv) Qt(P2, dv) (2.121) The consequence is that, if (XA) and (X2) are two independent copies of Dawson Watanabe superprocesses over (t) with branching mechanism T, where X0 = k1 and X02 = A2 then Xt = X' + X2 is a Dawson Watanabe superprocess, Xo = 1I + Z2. (2.122) Property 2.121 (and 2.122 as well) is a branching property, i.e., every Dawson Watanabe superprocess is a branching process. Let us fix Az E M+. Then, for every a > 0, ap E M+. The branching property 2.121 implies that a family {Qt(al, dv); a 0} (2.123) is a convolution semigroup (notice that t and p are fixed, and a E R+ is a parameter). It shows that, once we know Qt(6, dv) for every x E E, we can use 2.123 to find Qt(a6., dv), and then 2.121 to find Qt(I, dv) for every 1L of the form .= > aS, (2.124) i=l Let us also mention here that in the case of standard space (E, E) and correspond ing space (M+, MA), it has been proven in [18], pp.264265, that, if CQ,(f)  (f) as n * +oo, for every f E bps, then there exists a subprobability measure Q, such 66 that CQ = L, where denotes the Laplace transform. In particular, if (L,t, Vtf) (/i, Vtf), for every f E bpE, then CQt(m,dv) Qt(p,Idv) (2.125) CHAPTER 3 ONEDIMENSIONAL CASE In this chapter we describe the simplest possible superprocesses, i.e., the super processes over Markov chain with one state space. Since these superprocesses are onedimensional branching processes, they are recruited among processes which were studied before. We use onedimensional superprocesses then to describe general su perprocesses over deterministic Markov processes which stay in the initial state for ever (but the state space of the Markov process is arbitrary). In this way we obtain the "behaviour" of the superprocess which is "influenced" by the branching mecha nism part, only. Then, in the next chapter, we show how in some cases of Markov processes we can get a description of the corresponding superprocesses by using the results of this chapter. It should be emphasized that our main tool is the analysis of the supertransition function. 3.1 Supersemigroups and Infinitesimal Generators We restrict our attention to the onedimensional case in this and the following section. Hence, we consider the case when E =. {1}. Then a branching mechanism '! depends on A only, i.e., 1(A) =b cA2 + (1 ee")n(du) (3.1) where b E R, c E R+, and f+'" un(du) < +oo. In this case we do not have much choice for (st), i.e., (t = (0 = 1, Tt = Id,, and A = 0. Also M+ = R+, and JL and f are nonnegative real numbers. Using now the results from section 2.4. in this special case, we conclude that a DawsonWatanabe superprocess over (t) with branching 67 mechanism T is an R+valued Hunt process (Xt) whose infinitesimal generator g is given by ( cd2F bdF f+ (GF)(dP) = 2 c F b + [F( + u) F(p)]n(duj) (3.2) The supersemigroup (Qt(j, dv); / > 0) is a convolution semigroup with respect to 1L, whose Laplace transform is determined by (Vt; t > 0), where, for every f > 0, (Vf) = f+ f [b(Vff) c(Vf)2 + f (1 e(f)u)n(du) ds. (3.3) Since we know that Vtf exists, 3.3 shows that t Vtf is a continuous function, and, again by 3.3, t * Vtf is a cntinuosly differentiable function on R+, which satisfies the differential equation f d() = b(Vtf) c(Vtf)2 + fo+( i e(V'f))n(du) (Vof) f Let us consider some cases in which we can explicitly solve 3.4. Case T(A) = bA cA2; c > 0, b E R. In this case the differential equation 3.4 is a Riccati type equation Sy' = by cy2(3.5) y(0) = f 3 which can be easily solved, by separation of variables, to give f Vtf =ebt (3.6) e6' + fc ' where (ebt 1)/b is defined by ebt t { 1 etif bi 0 b (3.7) b limbo 1 = t if b = 0 Hence, in the case when b = 0 we have f Vtf (3.8) 1 +fct Using the fact that Vtf describes the Laplace transform of Qt((, dv), and by following simple computation (, Vf) = /) est + fCe bt c t eb1 b cX c_ f C+ we obtain that (recall that 4 and v are points in R+ here), " Fe^ Ga)+(dv) (3.9) n=I "1 Qt (, dv)=e z 6o(dv)+Z where ~ EXP eb (3.10) where, as usual, EXP(A) denotes the exponential distribution with parameter A > 0. The infinitesimal generator of this process is (by 3.2) d2 d c p2 by (3.11) If we compare these results to Corollary 2 of [44] we get the following simple conse quence: Corollary 3.1.1 (Xt) is a onedimensional superprocess with continuous paths if and only if it is a onedimensional continuous branching diffusion. In this case (Xt) has a transition probability given by 3.9 and infinitesimal generator given by 3.11. Remark 3.1.1 It should be noted that these processes have been studied before, and it was observed that they represent superprocesses. Here we have shown how these results can be derived in very simple and elementary way following our approach. Also, the transition function is not usually mentioned in the form 3.9 in the literature, and in the literature on superprocesses is usually not mentioned at all. However, in our line of approach we will rely heavily on a transition function. The diffusions with infinitesimal generator 3.11 were studied first by W. Feller in 1939 and especially in 1951 [26]. At that time continuous state branching processes were not introduced yet, and Feller has shown that these diffusions are the limits of discrete parameter branching processes. He treated them as (0, +oo)valued diffusions with 0 as an absorbing boundary. We include 0 in the state space, and because of that we have a singular first term in 3.9. Feller computed also a density part of the transition function 3.9 in terms of Bessel functions. The case c = 1/2, b = 0 is treated in [42] pp.100102, again from the diffusion point of view. In this case the transition function has a nice form Qt(I, dv) = e t o(dv) + 2e t 1(4 ) (3.12) t t V where I1 is the modified Bessel function of the first kind. We will not use 3.12 any more. It is mentioned here just to complete the description of the processes above. In 1958 M. Jirina introduced continuous state branching processes in [36]. In 1967 J. Lamperti covered the onedimensional case in [44] and [45]. He gave us formula 3.6 (for the first time as far as we can tell). A nice brief account of these processes from the point of view of branching processes is given in [1] pp.257261. Several authors on superprocesses, like D. Dawson, S. RoellyCoppoletta, etc., emphasized that these diffusions (or some of them) are special cases of superprocesses. Almost exclusively they deal with the infinitesimal generators. O Case k(A) = m(1 ea ); m > 0, a > 0. This is the case when n(du) = m8,. Again, the differential equation 3.4 is easy to solve, since it is in a form y'=m(1 e) (3.13) y(0) = f Separation of variables gives the solution Vtf = In [ea(eaf 1) (3.14) a Simple computation shows that e_(,vt = = em f[ ef(  [eamt(eaf 1) + 1]/a eMte [1+ e 1] Since eaf(eam 1) E (1, 1) we can apply Newton's binomial formula to get e(,'f) = ete~mt eaf(amt 1) = n=O n ( a ) eJ^mt/amt 1) enf(*+na) It follows that the supertransition function is given by Qt(, dv) = ( em't(eam' 1)"6,+,(dv) (3.15) n=0 The infinitesimal generator of this process is (by 3.2) (CF)(I) = m4[ F(I + a) F() ] (3.16) Remark 3.1.2 There are descriptions of these processes in [44] and [1], but they are treated mostly as timechanged Poisson processes (see more about it in the next section). However, we are not aware of formulas 3.14 and 3.15 in the literature. Here they follow easily from the general approach on superprocesses. Let us also mention that in the case a = m = 1, and = 1, the supertransition function 3.15 is given by Q(1l, dv) = E e'(1 et)"6i+n(dv) (3.17) n=0 which is a geometric distribution It is a good place to clarify certain facts about simple transformations of superpro cesses. Let (Xe) be a DawsonWatanabe superprocess over a Markov process ()) on (E, E) with branching mechanism W(x, A). We consider general LCCB space (E, E). Let f > 0 be a bounded, nonnegative, Emeasurable function. Then the stochastic process (Yt), defined by Yt = (Xt,f) (3.18) is an R+valued stochastic process. One may hope that (Yt) is a onedimensional superprocess somehow related to 9. This is particularly so in the case when f = 1, i.e., (Yt) is the, so called, totalmass process. However, this is not always the case. The following two examples indicate typical difficulties. Example 3.1.1 Let ((t) be the uniform motion to the right on the real line, and T(x, A) = A. Let f = 111,+0). Using Theorem 2.3.1 and formula 2.85 we obtain Y = (X, f) = XXo(dx) E [f(t)e foI()du = Xo(dx) (ef(x + t)) = e' Xo([l t, +oo)) Hence, for every Xo =p_ / 0, where supp(/p) C (oo, 0), we will have that Yt = 0, for 0 < t < 1, and Yt > 0, for t large enough. But, there is no onedimensional superprocess with such a behaviour. For every onedimensional superprocess it is true that if it starts at zero, it stays there forever, since the Laplace transform of Qt(O, dv) is identically one.O Example 3.1.2 Let (t) be again the uniform motion to the right on the real line, and k (x, A)= b(x)A, where 1; x <.1 b(x)= xi ; x (1,1) 1; x>l 1 Let f 1. Using Theorem 2.3.1 and formula 2.85 again, we obtain Yt = (X,,1) = (Xo, E [e f'b()d . If Xo =8 a R, then Yt = E" e fo b(])du = e b(a+u)du It shows that Yt is a deterministic process, and for a = 0 it is equal to { et2/2 ; t [0, ] (3.19) y e e1/2 + (t ) ; t> >191 However, there is no such onedimensional superprocess. As we can see from 3.2 the only deterministic onedimensional superprocesses are the processes of the form Xt = Xoebt.O Notice that in the first example the coefficients of T are constant, but f $ 1, while in the second the coefficients of q are not constant. Several authors considered constant coefficients and f 1 (see, for example [52]). In this case (Xt, 1) "behaves" as we would initially expect. For the sake of completeness let us prove this result. Proposition 3.1.1 Let (E, E) be a standard space, and (Xt) a DawsonWatanabe su perprocess over ((t) with state space (E, E), and with branching mechanism T1, where T has constant coefficients, i.e., b(x) = b, c(x) = c, and n(z) n. Then (Xt, 1) is a onedimensional superprocess with the same branching mechanism T. Proof. Notice that the statement makes sense, since 'I can be regarded as a branching mechanism over {1}, as well. Let (Vt) be a nonlinear semigroup which corresponds to (Xt). We claim that for every A > 0, Vt(Al)(x) is a constant function. Recall, by construction, that Vt is obtained as a limit of V,"+(Al)(x) = T(Al)(X) + T_.,[WP(Vn(A))](x)ds Jo where Vo = 0, and Tt(A1) = ATtl A. Assuming that V,"(A1) is a constant, we get that I(V,"(A1)) is a constant function, since T has constant coefficients. Then Tt_.,[(Vn(A1))] is a constant, too, since Tt,l = 1. It follows that V"+1(A1) is a constant function. By mathematical induction we obtain that x i Vt(Al)(x) is a constant function, for every A > 0, and it satisfies Vt(Al) = A + + '[V8(Al)]ds If we define WtA = Vt(Al), then Wt satisfies 3.3 (by the equality above). Since, for A > 0 and Xo = L, I+ eA(x,1)dP = M e(XtAl)dP = e(,u'v(A)) = e((E).Vt(Al) e(Xo,1)'Vt( ) e(Xo,1)'Wt we have shown that (Xe, 1) is a onedimensional superprocess with branching mech anism T. Q.E.D. Remark 3.1.3 It is possible to study (Xt, f) also by using martingale theory approach. We will not enter this subject. Let us only mention some references like [10] and [52]. Several results on semimartingale properties of exp((X, f)) , where f is in the domain of the infinitesimal generator of (4e), are given in Chapter 5 of [10].E 3.2 Time Change and ZeroOne Law We have seen in the previous section that onedimensional superprocesses are actually onedimensional branching processes studied by J. Lamperti in the late six ties. J. Lamperti has also shown that these processes can be obtained as a time change of Levy processes. Here we will emphasize Lamperti's result and show that related to this is an interesting zeroone law for Brownian motion, which in some sense complement EngelbertSchmidt zeroone law (see [23]). Let k(A) be given by 3.1, and (L L) a corresponding Levy process given by 2.76. Then we have Lf = bt + VcBt + Jt (3.20) where (Bt) and (Jr) are independent copies of standard Brownian motion and pure jump process with positive jumps, respectively (see 2.76 and [35] for details). In particular, (Lf) is a strong Markov process with the infinitesimal generator given by d d2 r0+m b +cd + +[ F(x+ u) F(x)] n(du) (3.21) dx dx2 Jo Consider now only the cases when L =. a > 0, and let (Ltf) be the process (Lt) stopped when it reaches zero. Then (L" ) is a Markov process with state space R+, whose infinitesimal generator is given by 3.21 for positive x, and is equal to zero for x = 0. Consider the additive functional (A*) defined by A? = f (3.22) Notice that (A") is a continuous additive functional, except maybe at the point o = inf{t > 0 : f = 0 }, which is equal to inf{t > 0 : L = 0 }. Obviously AZ+ = +oo, but it is possible that A*_ < +oo. We will see later that it is so indeed in many cases. Therefore, we define A' to be equal to A,0_. Hence, t A' is continuous and strictly increasing on [0, T], and the same is true for its inverse. We define the inverse of t A* to be equal to T0, for every s > A*. Once this is settled the standard results on timechanged process apply. Therefore, let (Xt) be a process obtained from (L'*) by a random time change, corresponding to the additive functional (A'). Using Theorem 10.12 from [16] we obtain that (X/) is a Markov process whose infinitesimal generator is given by 3.2, i.e., (Xlt) is a onedimensional superprocess described in section 3.1. Remark 3.2.1 The result above belongs to J. Lamperti. It was announced in [44], but as far as we know its proof was not published in author's later papers. Of course, the proof is more or less standard application of random time change, and it was outlined in [44]. One only has to be more careful with the possible discontinuity at 70, as we explained above. O Let us investigate some cases in which A"_ is finite. Consider the standard Brownian motion (Bt) on the real line. Let 70 = inf{t > 0 : Bt = 0 }. A consequence of the following theorem is that A_ is finite if T(A) = A2. Theorem 3.2.1 (ZeroOne Law) For every a > 0 a [ ds < oD ; if 0 < a <2 P"[J B< B < 0 ifa>2 Remark 3.2.2 Notice that Theorem 3.2.1 is an interesting result in itself. Recall that the EngelbertSchmidt 0 1 law implies (see [23]) po t ds < + ;Vt >0 1 if0 o IB" 1B;1I = 0 ; if a > 1 which says something about the behaviour of the Brownian trajectory when it leaves zero (or on the right hand side of zero). Our result 3.23 complements Engelbert Schmidt's result in the sense that it compares what is happening when Brownian motion enters zero (or on the left hand side of zero). Obviously, the behaviour is different. Notice also that our result is not the consequence of general criteria in [23], since these result only show that for a > 1, and a > 0, P1. I < +oo;Vt > 0 =0 which does not imply anything about the integral up to 70.3 Proof. Let Tb =inf{t > 0 : Bt = b} and, for n > 1, an = a/n. Then we have, since pa(Bo = a) = 1, o ds "=+i ds. Jo d I pa a.e., a n=1 Ba'n and the sum on the right hand side is the sum of nonnegative, independent (with respect to P) random variables. Hence, it is P" a.e. finite or P" a.e. infinite. Consider 0 < a < 2 first. If s < ,,Tl then B, > an+I P a.e. It follows that s < (1 7,+ n) pa a.e. o Ba n=1 an+1 If we denote (1/a )( ,+ ra, ) by Zn, then { Z,; n > 1 } is the sequence of non negative independent random variables. Applying Kolmogorov three series theorem we obtain that Pa 1 Zn < +00 = 1 , n=l 1 if ^P (Z,> 1) <+oo , and ZE [Zn z Recall that P" distribution of Zn is equal to Po distribution of (1/aI+1).Tr/(n(n+l)). It follows that pa (Z +00 n(n^e) n^ Z> =(n) e 2s ds < P"(Z,,> 1)= C +;), 2 e _"27rS a + ds 2a (n + 1)"/2 Sv27n(n + ) J() s/2 v n(n + 1) aa/2 and the expectation E Z8 (. ] = a s + e 2. ds < a(n + 1)" S d = i2 a(n + 1)' a'12 2n(n + 1)a s3/2 s V n(n + 1)a (n + 1)F/2 Since a < 2, then a/2 = 1 e, for some c > 0, and 0 (n + 1)1' "0 1 : = <+ 00 n=l n(n+ 1) += n(n +l)" Hence, the two series mentioned above are convergent, and we finished the proof of the first part of the theorem. To prove the second part, it is enough to prove that ro( ) ds B() 00 for w E F, where Pa(F) > 0. Using the wellknown L6vy's modulus of continuity P" limsup sup I B, 1 = 1 , o o_<, and the fact that pa( ro < 1) > 0, we can choose F to be the set of such w that ro(w) < 1, Bo(w) = a, and there exists 8(w) > 0, such that for every e < 6 sup B B <2 O If t2 = r7o() and tj = t2 e, then it follows, since Bt,(w) = 0, that, for every e < 8, ()_ <2(w) 2 /2e ln(l/e) Hence, we obtain *(o) ds o(W) ds B(Wj) (w)S(W) B 1 f( ) de 1 S cn(/e) [ ln(ln(l/e))] I(= +00 for every w E F. Q.E.D. With these observations we finished our description of onedimensional superpro cesses. Our idea is to use these results in describing more general cases. First we will describe (Idt, ))supersemigroups over the general space (E, E). To do so we have to introduce some special measures on the space of measures on (E, ). This is the content of the following section. 3.3 Measures on Rays Let (E, ) be a measurable space, and M the space of all finite signed measures equipped with aalgebra M, which is the smallest aalgebra such that, for every f bounded Emeasurable, the mapping pt * (tz, f) is measurable. M+ is the cone of (positive) finite measures, and M+ = M+ n M. As before, bE (bpE) denote the set of bounded (bounded and positive) Emeasurable realvalued functions. Consider t,(da), a positive kernel from (E, ) to (R+, B(IR+)), i.e., for every G B(R,), x i ( t(G) is Emeasurable, and, for every x E E, t,(da) is a afinite (positive) measure on (R+,B(R+)). Assume also that for every compact set K C R+, x t,(K) is bounded. Using t,(da) we define i : E x M+  [0, +oo] by i(2, N) = t{ a E R+ I a, E N } (3.24) for every x E E, N E M+, where S, E M+ is the point mass at x. Assuming that 9 is a kernel, we can define, for every P E M+, a measure 6, on (M+,.M+) by (N) = (x, N) (dx) (3.25) The following proposition shows that this way we obtain a positive afinite measure on (M+,M+). Proposition 3.3.1 1 is a positive kernel, and E, is a positive afinite measure on (M+,M+). For every F : M+ [oo,+oo], which is either positive M+ measurable or bounded M+measurable, I F(v)8(dv) = f F(a6,) t(da) )(dx) (3.26) Proof. Consider the function H : R+ x E + M+ given by H(a, x) = a6, We claim that H is B(R+) C/M+measurable. It is enough to prove that, for every f E bps, (H, f)(a, x) = a f(x) is B(R+) 0 EC/B(R+)measurable, which is obviously true. Fix N E M Then 9(x, N) = t,{a E R+ (a, x) E H1(N) }, i.e., Si 9(x, N) = JR+ lH(N)(a,x)t,(da) , which is E+measurable, since t,(da) is a kernel and H'(N) E B(R+) The fact that i(x, ) is a positive ufinite measure follows directly from 3.24 and properties of tx(da). It implies immediately that O, is welldefined positive measure on (M+, M+). To prove that O, is ufinite, let us consider a sequence Nk = {v E M+ I v(E) < k }, k E N. Obviously Nk E M+, and (Nk; k > 1) covers M+. Notice that )(x, Nk)= t([0, k]) . Boundedness of x t,( [0, k]) implies that Q,(Nk) is finite (recall that p is a finite measure). Let F = 1N, N E M+. Since i9 is a kernel we obtain S1(u),(d ) = J 4(x, N) p(dx) = = J 0 lN(abx) t,(da) P(dx) JJo Standard measuretheoretic argument shows that 3.26 is valid for any F, which is either positive M+measurable or bounded M+measurable. Q.E.D. Definition 3.3.1 Measure e,, defined by 3.25, is called a measure on rays with pa rameter iL, and with respect to a kernel t,(da). Notice that 3.26 applied to F(v) = exp( (v, f) ), f E bp', gives the formula for the Laplace transform in the form M+ e^vf) ,(d) = +0 e"6' t,(da) p(dx) (3.27) We will be interested in some special measures on rays. Let A(x, N) be given by 3.24, where tx(da) is Lebesgue measure, for every x E E. Then the corresponding measure on rays, A,(N)= A(x, N),(dx) (3.28) is called the Lebesgue measure on rays. Remark 3.3.1 In the case E = 1 }, A, is a multiple of the standard Lebesgue measure on R+. However, already for E = {1,...,n}, n > 2, A,, is a measure which "lives" only on the coordinate axes. Similarly, if E is a topological space, A, "lives" only on rays { a6x : a > 0 } through 65, where x E supp(4).0I Assuming usual conventions 1/ + oo = 0, 1/0 = +oo, 0 +oo = 0, we obtain the Laplace transform of A, in the following form, J + d d J (dx) Se A,(d = e da (dx)= (3.29) M+ JE0 JE f(x) where f E bpE. Using A, we define the exponential measure on rays exp,(h), where h E bpE, given by exp,(h)(dv) = e("'hA,(dv) . (3.30) Obviously exp,(h) is absolutely continuous with respect to A,, and its Laplace trans form is given by JM+ e("f)exp,(h)(dv) = fM e') e^(h)A,(dv) = S (dx) (3.31) ~JEf(x) + h(x) 3.4 (Idt, )Supersemigroups Let (E, E) be a standard space and (M+, M+) the measurable space of all finite (positive) measures, defined as in the previous section. Let I'(x, A) : E x R+  R be a branching mechanism, defined by 2.55. If we consider such a Markov process ((t) on E, that 6t = 6o, then the corresponding semigroup (Tt) is actually (Idt). As we have seen in section 2.2., there exists a unique (Idt, ')supersemigroup, de noted by Q' (p, dv), and determined by its Laplace transform. We will denote the corresponding nonlinear semigroup by (Vt') and (Idt, T)superprocess by (X,). The purpose of this section is to describe Qt(y, dv). We will use onedimensional superprocesses and supersemigroups in this description. After that, in the next chap ter, we will use (Idt, t)supersemigroups as "building blocks" in the description of the general case of superprocesses. Recall the notation from section 2.3., where we denoted the point mass at x E E by 6x, and the point mass at fi E M+ by A,, i.e., AA is an example of a probability measure on (M+, M+). Notice that for every x E E, A * Q(x, A) is a branching mechanism on the onepoint set {1}. We denote it by T,. Then the corresponding onedimensional superprocess, denoted by ( f"), is the timechanged process of the L6vy process (LZ') (see 3.20) stopped at zero. We denote the transition probability of (e*') by qt,,(a, db), and corresponding nonlinear semigroup (which is actually a function in onedimensional case) by vt,x(a). Then vt,, : R+  R+ is a function which satisfies the integral equation vt,.(a) = a + (x, v,, (a))ds (3.32) or, equivalently, the differential equation of (v,,(a) = '(x, vt,.(a)) (333) vo,(a) = a The Laplace transform of qt,=(a, db) is given by 0 ecqt,x(a, db) = eav .(c) (3.34) Consider (Vt~). Recall that, for every f E bpE, (Vt'f)(x) = f(x) + f P(x,(V f)(x))ds (3.35) The uniqueness of the solution of 3.35, and of 3.32, shows that, for every f E bp$ and x E E, (V, f)() = vt,,(f(x)) (3.36) which implies immediately that IM e"')Qf(L, dv) = exp ( vE ,(f(x))(d) (3.37) In the special case when 1 = m 6,, m E R+, z E E, we obtain IM e<.'f)Q(m6s,,dv) = em"",,(f())= JM+ = e f(z)bqt,z(m, db) (3.38) which proves the following simple result. Proposition 3.4.1 If (X) is a (Idt, ')superprocess with Xo = m6z, then X,' = . 6. (3.39) where I = m. Remark 3.4.1 If we think of superprocesses as the distribution of a population of particles, then 3.39 says that, if the entire population has mass m and is positioned at z, then at the moment t, the entire population is still at z, but its mass changes with the law of onedimensional branching process which we described in sections 3.1. and 3.2. Notice that (t) in this case is trivial, i.e., does not change the starting position z. We will see later that this, intuitively very acceptable behaviour (where ((t) affects the position of the population independently from 9, which affects only the mass of the population), appears in the cases where (6t) is "simple". However, in general case the behaviour is more complicated and is not clear.0 Formula 3.38 can be used to get Qf(pt, dv) for some other 1. The idea is to use the branching property 2.121, and limits of the type 2.125, to calculate Q"(p,, dv). We can not obtain explicit formulas by this procedure in the most general case. However, in some special cases we can use measures on rays (see section 3.3) to obtain nice explicit formulas for Q (p,, dv). The procedure is described below. Assume that, for every t > 0, there is a positive kernel Tt,,(da) from (E, E) to (R+, B(R+)), such that x i Tt,,(K) is bounded for every compact K, and, that there is a function ht : E  R+, such that vt,X(c) = ecbTt,(db) h (3.40) Using the results from section 3.3 we obtain that, for every t > 0, there is a measure on rays OE,t such that its Laplace transform is JM+ e("f)O ,t(dv) = 1E j+o ebf()Tt,z(db)p(d) = = (, ,vt,((f(x))) + (,ht(x)) (3.41) Recall now that the Laplace transform of the convolution of two measures on the space (M+, M+) is the product of their Laplace transforms (see section 2.5). Then 3.41 and 3.37 imply that Q(, dv) = e^,h )). ,( (3.42) n=1 n Let us consider now particular branching mechanisms and show how we can apply results obtained in this section. Case '(x,A) = b(x) c(s)A'; b bE, cE bp, c > 0 Using 3.36 and 3.6 we obtain, for every f E bpf, (V,~ f)(x) = f(x) (3.43) e) + f()C()c3)43) where the quotient (eb()t 1)/b(x) is defined by 3.7. This is the example which satisfies 3.42. Let O,,t ~ exp,,(g), where exp,(g) is defined by 3.30, and 77 and g are given by eb(x)t eb(x)t (dx) = cb)t1 2(d) and g(x) = C eb )1 (3.44) b() bc( x) Let a function ht be given by 1 ht(x) = c(1. (3.45) C(X) b(x) Using 3.31 and 3.41, we obtain that Q (it, dv) is given by 3.42, since (, ht(x)) + JM e()O t(dv) = eb( )t = j(d x)) b(x) ( j \) S() b( (c(x)e j) (f()c(x) b()) + eb(=)) E eb()t+ f (Xeb() ^ (x) {d b(x) J~^ f(~ x}^ *,tib5 Simple computation shows that in the language of random measures O,,t is the L6vy (canonical) measure of Q"f(, dv) (see [37], chapter 6). Notice that, when c(x) = 0, then (Vtf)(x) = f(x)eb(z)t (3.46) and Qt (p, dv) = Ae(.).t(d)(dv) (3.47) which is a special case of 2.86. Case %(z, A) = n(x)( 1 e"()^ ); n, a E bp,, a > 0 Notice that for every a > 0, n > 0, real numbers, it is not possible that there exists a measure T on R+ such that SeAbT(db) = In [e(ea 1) + 1h , 0 a since A = 0 would imply T(R+) = h < +oo, and A  +00 would imply T({0}) = +oo. Therefore, in this case we can only use formulas 3.36 and 3.38 to get (Vtf)(z) = 1a ln [en()a(0(e () 1) + a(x) (3.48) and Qf(mS, dv) = .() emn(2)t (en(z)(z)t 1) A(+ )),(dv) (3.49) k=o k Let us also mention the case of E = {1,...',n}. In that case T is determined by 1,..., 2, and every p E M+ = R+ is = (/i,... ,n). Hence, if we denote Qt (lzi8{, dv) by Q'(Li,dv), then Q?(P, dv) = Q(j~i, dv) ... Q(n, dv) (3.50) Therefore, the (Idt, I)superprocess (Xt) which starts at y E R' is given by X, =(e ,...,I , (3.51) 87 where (d4') are independent copies of onedimensional Iisuperprocesses, such that (Lt') starts at pi E R+. The infinitesimal generator 9 of (Xl) is given by 2.115, i.e., (f)(x) = E c bi + [f(x + eiu) f(x)]ni(du) (3.52) Notice that one of the characteristics of the process (Xt) is that for almost all trajectories w (X )(o) = 0 = (X,+);(w) = 0, Vh > 0. (3.53) We will use this observation later. CHAPTER 4 TRANSITION PROBABILITIES OF SUPERPROCESSES We will apply results from Chapter 3, to describe supersemigroups of general superprocesses. In the case of "simple" Markov processes, like the uniform motion to the right we obtain explicit formulas. However, in more general cases we are not able to say much. Already in the case of Markov chains we can describe supersemigroups only locally with the application of CameronMartin formula. 4.1 TrotterKato Formula In this section we restrict ourselves to the assumptions of section 2.4, i.e., E is a locally compact Hausdorff space with a countable base, EA the onepoint compact ification of E, (t) a Markov process with state space (E, B(E)) (A is a cemetery), whose semigroup (Tt) satisfies the Feller property, and a branching mechanism T that satisfies 2.93 and 2.94. We have seen, in section 2.4, that under these assumptions the superprocess (Xe) is a Feller process with state space M', and (Xt) actually "lives" on M+. We denoted its infinitesimal generator by g, and 2.111 shows how g is defined on its core Exp(M TD(A)) (see 2.109), where A is the infinitesimal generator of (Tt) extended to C(Ea) (see 2.92). However, in section 3.4 we described (Idt, I)superprocess for every standard space (E, $). In particular, we have the description of (Idt, 1I)superprocess under the assumption of this section. Let us denote (Idt, ')superprocess by (X*t), and its supersemigroup by (Q(). Of course, (X)) is also a Feller process on M+ and Exp(M+, D(0)) is a core for the infinitesimal generator of (X,). Since D(0) = C(EA), it follows that for every A, Exp(MA, D(A)) C Exp(MA, )(0)) = Exp(M+, C(EL)) Let us denote the infinitesimal generator of (Xt) by 9'. Then, by 2.111, for every f E C(Ea)+ and pt E MA, ["^(F)] (p) = [F,(p)] (p, ~f) (4.1) Consider now the case when _= 0, i.e., the case of (Tt, 0)superprocess. Following the notation of the section 2.4 we have that i 0, and Vt = Tt. Since Tt is linear and positive we have that pTt, defined by (tT,, f) = (', Tf) (4.2) is a measure in M+. Therefore, we have an explicit formula for (it, 0)supersemigroup. If we denote (t, 0)supersemigroup by (Qf) and (Tr, 0)superprocess by (Xt), we ob tain that Qt(A,,dv) = A't,(dv) (4.3) and Xt = Xtt (4.4) Let us denote the infinitesimal generator of (Qf) by G9. Again, section 2.4 shows that Exp(M+, D(Ai)) is a core for GE, and, for every f E C(Ea)+ n D(A), f strictly positive, and 1 E M ,A [ (F,) () = [F,(2)] (, f) .A (4.5) Using 4.1, 4.5, and 2.111, we obtain that g = g' + gt (4.6) on Exp(M 2D(A)). Since we have the description of (Q ) (section 3.4), and the description of (QS) is simple (see 4.3), the equality 4.6 indicates that we can use the wellknown consequence of TrotterKato formula (see, for example, [47], Corollary 5.5) to obtain an approximation formula for (Qt). Theorem 4.1.1 For every F E Co(M+), and, for every t > 0, QtF = lim Q' F=lim (Q Q/ n F (4.7) and the limit is uniform on bounded t intervals. Proof. The proof is now almost straightforward application of Corollary 5.5 in [47]. We only have to be careful about the fact that we are dealing with cores for infinitesimal generators, and not with their domains. All three semigroups (Qt), (Q?), and (Qi) are Co semigroups on Banach space Co(MA), and all three are contractions. Since D(g~) n D(ge) contains Exp(M+, D(A)), it is dense in Co(MA ). The image of Exp(M+, D(A)) with respect to the operator zI g, where z is a complex number, such that Re(z) > 0, is dense in Co(M+), since Exp(M+, D(A)) is the core for g. By 4.6 the same is true with respect to the operator zI (9" + 9f). It follows that the closure of G9 + G is the infinitesimal generator of a contraction Co semigroup, which is given by limit in 4.7. By 4.6 and 2.109 we obtain that D(a) C n(Gd + C) and (4.8) However, both g and g* + ge are infinitesimal generators for Co semigroups of con tractions. By 4.8 this is possible only if Q = G + Qg. Q.E.D. We have explicit formulas for (Qf~) for several cases (see section 3.4), and for (Qf) always. In particular, (Qf) has a very simple description (see 4.3). One may hope to find explicit formulas for (Qe), by using Theorem above. Unfortunately, it does not seem to be so. The following example illustrates how complex these formulas can be in general. In some special cases we will obtain nice explicit formulas in section 4.3. Example 4.1.1 Let (t) be a twostate Markov chain, which stays for exponential time in state 1, then jumps to state 2 and stays there forever. More precisely, its semigrouop is e' 1 et T = 0 1 (4.9) and its infinitesimal generator is A= ' (4.10) Of course, in this case E = E = {1, 2}, and C(EA)= R2, and M' = R_. Let the branching mechanism be T(x, A) = A2, i.e., it has constant coefficients. Then the (Tt, P)superprocess (Xt) is an R valued stochastic process, whose in finitesimal generator is given by (see 2.115), 82 02 0 0 x jl + x2  X1 + X1 (4.11) 1 OX2 1X 1 O2 In this case we have explicit formulas for (Qf) and (Qf) (we do not need "tilde" notation here, since E = Ea). Moreover, we can find also an explicit formula for a nonlinear semigroup (Vt), which is the solution of 2.61. However, even in this (relatively simple) case we were not able to extract explicit formula for (Qt). Let us compute the explicit formula for (Vt). For every f = (fl, f2) in C(EE)+ = R+, Vtf = ((Vf), (Vtf)2) is also in R_ We claim that Vtf is given by (Vtf) f2 + (fi f2)et (4. f2t + 1 (f2t + 1)2 [ft Id. f 1 (>,)()1 = + 1((4)12) f2 (f2+1)2 (fl and (f)2 + (4.13) f2t + 1 Using results from section 2.1, it is enough to check that Vtf is welldefined, positive, and that it satisfies 2.61. There are no problems with the second coordinate 4.13, since this problem is solved already in section 3.1 (see formula 3.8). Thus, we will concentrate on 4.12. If fi f2 > 0, then (Vtf)x is obviously welldefined and positive. Consider the case when fi f2 < 0. We define a realvalued function h : R+  R by ft e'ds h(t)= (feds + f2) + 1 fi (f2S + 1)2 Since fi f2 < 0 we obtain, for every t > 0, ds 1 1 h(t) > (s (f f2) +1= 1 i (f f2)+1 (f2s + 1)2 f2 f2t + I t fit + 1 =^ (12)+1= >o . f2t +1 f2t + 1 Therefore, (Vtf)i is welldefined. Let us prove that it is positive. By the proof above we obtain et(f2t + 1)2. h(t) 2 et(f2t + )2 1 > f2t+ 1 > (f2t + 1)(ft + 1) > 0, which implies, since fi f2 < 0, (Vtf > 2 f f2 f2t + 1 (f2t + 1)(fit + 1) fi + ff2t = >0 (ft + 1)(fit + 1) It remains to be proved that (Vt) satisfies 2.61. Let us define a realvalued function H : R+ Rby 1 H(t) = (f2t + 1)2 h(t) By differentiating H(t), and using that H(0) = 1, we obtain H(t) = 1 H(s) e(f, f2)H(s) +2 f2 ds JO f2S + 1 Multiplying both sides by et(f f2), and then adding f2/(f2t + 1) to both sides, and using the fact that (Ttf)i ft f2 ds f2 + (f f2) .o if2s + 1 f2t + 1 we obtain (Vtf)i = (Ttf)i t f 2 ds 0 f2s + 1 Ie` e [e'(fi f2)H(s)]2 + 2 e'(f, f2)H(s) ds . By changing the variable of integration, and by writing 1 = (1 e') + e', we get (Vf)' = (Tf) [T,(V_,f)2] , which is exactly 2.61 for the first coordinate in this case. As we have just seen it was lengthy, but elementary, to check that 4.12 and 4.13 give the solution of 2.61 for our (Tt) and l. However, it was not so easy to find 4.12. To do so we were trying to solve the corresponding differential equation. Combination of several useful suggestions from [38], and many guesses led us to 4.12. Using results from section 3.4 and formula 4.3 we obtain (4.14) Qf((Al,2); (d 2))= A(A .,, (1_ec)+,)(dvi, dV2) , and t '((l, 2); (dvi, dV2)) = (dvi, d2) e (4.15) (0 ) n / k ] nk 1 n vt) l k ev/t ,( 2(nk) le(/t n=O n! =0(k) (n k) where r is the wellknown gammafunction. These results imply the following formula for the mth term in TrotterKato ap proximation formula 4.7 [ e(dvi dv t/e m [Q,/mQ/m] ((pl, #2); (dv, dv2)) = (dv, dv2) .e t/ V1 '2 Ae nt/1m) = [klet/m]' 0[1(1 et/') + I2 ]' n [1e ^/ nk t /m 2 ne=0 n2!(t/m)2 n, k= 0 ( (t/2m)njn +n ( t2 ) k2 1 n2k2. f=2O fl2!(t/m)2n2 k2kO nzkz .22 ( n2 k2 1 12=0 12 (1 et/m)n2k1 nim=0 km=0 nmk nm, kIm 1 *==o Im (1 e'/")"=km" F(k1 + k2 + 12) P(ni + n2 (kI + k2 + 12)) r(ki)Pr(n ki)Fr(k2)r(n2 k2) 00 (t/2m )n _l+n ( ) \2(1  E ni!r(t/m)2nm 2 [" Eietl 2( r(k^_ + km + m) r(nni + nm (kin + k + m)) r(km )r(nm km) Despite the fact that the formula above seems to show some sort of "regularity", and that we have explicit formula for (Vt), we were not able to recognize if there is an explicit formula for (Qt). This question remains open.O 4.2 Superprocesses over Markov Chains Let ((e) be a Markov chain with state space {1,2,...,n}. It is determined by its semigroup (Te), i.e., by the infinitesimal generator A = (aij)". As we have seen already in section 2.4 and 2.5 the superprocesses over (t) are branching Markov pro cesses on R+ whose infinitesimal generators are given in Theorem 2.4.3. The class of branching Markov processes on R' has been studied by Watanabe in [57] (the paper actually deals with the case n = 2, but only to make notation simpler) and by Rhyzhov and Skorokhod in [51]. Of course, at that time (1969) they did not con sider the terminology of superprocesses. However, in their papers they characterized completely the infinitesimal generators of branching Markov processes on R" (see, for example, [57], Theorem 2, pp.449450). Comparing these formulas to 2.115 we can see that the class of superprocesses over Markov chains is smaller than the class of branching Markov processes (there are no termination coefficients in 2.115 and drift coefficients are restricted by b and A). Even more general class, of branching processes on R+ with immigration, has been studied in [40]. Following the notation of the previous section, we obtain from 2.115 that (Id,, ') superprocess (X') has the infinitesimal generator 9', given by r 2 F F o00 ('F)(p) = 2E c, bb + [F( + eu) F(I)]n,(du) (4.16) In particular, it shows that (Xt') is equal to the process (see 3.51 and 3.52) ( t t ,..., .) (4.17) where (d"') are independent onedimensional superprocesses with respect to ',(A) = %P(i, A); i = 1,... ,n. Recall (chapter 3) that each (T') is the timechanged process of the L6vy process (LP') stopped at zero (see 3.20). In particular, it shows that, for 