Ron Getoor, 1967-1995

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Title:
Ron Getoor, 1967-1995
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Language:
English
Creator:
Getoor, Ron
Chung, Kai Lai
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Folder: Ron Getoor, 1967-1995

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Subjects / Keywords:
Mathematics -- History -- 20th century

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University of Florida
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Full Text



November 30
Dear Getoors
Many thanks for your reply. I thought I understood your explanation of the
consistency of the two definitions until I recalled Henry's letter, which I there-
fore inclose for comments. Please return same*

At present I am forgetting about Willie's first paper and trying to make a
quick dive into his second to see if I want to continue the effort. I was again
stalled and, on the strength of Henry's last paragraph, beg to trouble you once more
about two questions.
first
p. 33 of second paper, can you explain thetequlity of (8.13)and the rest of
the computation up to the derivation of (8.15). Is x an arbitrary vector? This
must be a trivial thing but I just lost courage in spotting all the misprints.

p. 37, (9.7). This is more serious, why the first equation? If one substitutes
(9.6) into (9.5) one gets a term .CA(w). As I understand it this is the wth component
of the vector JICA as yet "unaquainted" (cf. p. 37). Well I suppose it must be a linear
combination of the CA(i)'s and CA(w); call it ZbiCA(r)+bwCA(w). Now the equation (9.5)
becomes

AA(w)bi-0(i)) yi+ ... yw=0 where yi=(k), as CA (w)

In order tht the range does not depend on A the above equation in the y's must have
proportional coefficients as A varies. I don't see why A (w) must be constant.

I do hope these rash thoughts do not disturb the uniqueness of the form of the
solution there.








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STANFORD UNIVERSITY
STANFORD, CALIFORNIA
DEPARTMENT OF MATHEMATICS
Jan. 19, 1967
Dear Ron,

Many thanks for the reply to my questions. I am not going
to Houston. Among other reasons McKean offered to come and talk
on his boundary paper here beginning c. 23rd.

May I trouble you again about some questions which I could not
easily dig out of Dynkin (perhpas all answered in your book). To
save you time I will parcel them out and leave blanks for you to
jot down the answers.

(1) Dynkin's definition of standard process .does not require
left limits for samples. Quasi-left continuity only implies conti-
nuity at each fixed t (together with right continuity of samples).
How does he get away without the left limits?





(2) Por a Hunt process where left limits are assumed, can any of
these be oo (the added point for compactification, is it the same as
your delta?) Does the ift limt exist as t ->a)? and can it be oo?






(3) For a standard process with left limits, can any of them be
oD at a finite time, and can it be oo as t tends to oo? Are a. a.
samples bounded in [0, t(w)] for each t(w)




(4) Using Hunt's notation except for prl if u is excessive is
E-> r TE a "capacity" in some sense? Please specify the sense and give refere
ence to Meyer if possible. If yes, which of Hunt's theorem imply it?







With best regards,





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UNIVERSITY OF CALIFORNIA, SAN DIEGO


BERKELEY DAVIS IRVINE LOS ANGELES RIVERSIDE SAN DIEGO SAN FRANCISCO 0 SANTA BARBARA SANTA CRUZ


DEPARTMENT OF MATHEMATICS POST OFFICE BOX I09
LA JOLLA, CALIFORNIA 92037

February 4, 1974





Professor K. L. Chung
Department of Mathematics
Stanford University
Stanford, California 94305

Dear Kai Lai,

Thank you for your recent letter. You are correct in assuming that
I exercised my will power. I had been planning to go to San Francisco
and only changed my mind the day before the meeting.

I have now had a chance to look at the 1968 paper of Dynkin that you
mentioned. I, too, was unaware of this paper. There is no doubt that
the basic technique of projecting (or balayaging) the functional o h(X )ds
onto the set V in Dynkin's notation (F in ours) is explicitly contained
in it. He then uses this to express the resolvent of the full process in
terms of the resolvent of the killed process. I only wish that you or one
of the other forty people to whom I sent preprints had called his paper to
our attention before our paper had appeared. However, this is definitely
my fault and Dynkin's priority must be acknowledged in print. I shall add
a note to our paper "Balayage and Multiplicative Functionals" acknowledging
his priority.

However, all this being said, I feel that it is not quite fair to say
that it is all in Dynkin except for "Strasbourg frills" for the following
reasons.

1) Dynkin does not invert the Laplace transform to obtain the relation-
ship between the semigroups for each t. As you know going from the almost
everywhere statement that is immediate from the Laplace transform to an
everywhere statement is not always trivial. In fact, in Getoor-Sharpe this
inversion is perhaps the trickiest thing.

2) Dynkin assumes that the set V is closed and V-Vr is polar.
The extension to general V is by no means immediate. The conditions on
V are precisely those sufficient conditions given our book that enable
one to assert that PV 1 is the a- potential of a continuous additive
functional. Moreover, Meyer goes a step further considering a homogeneous
Markov set and replacing the assumption of a standard process by a "right"
process.





Professor K. L. Chung
February 4, 1974
Page 2







3) I don't believe that one can use Dynkin's results directly to
obtain the conditional distributions given (Lt ) and ;(L t-). Nor is
it even implicit in Dynkin that there are two decompositions corresponding
essentially to X(L t) and X(L -).

Thus, in hindsight, it is easy for me to see our basic idea in Dynkin,
but I doubt very much that if I had read his paper in 1968 that I would
really have understood everything about last exit decompositions and distri-
butions. Moreover, I certainly would have published our paper even if I
had known of Dynkin's since I feel that it goes beyond his in a fundamental
way. The thing that I feel bad about is not giving him his full credit
for first using the balayage technique. I hope that I can correct this
in the near future.

Best regards,



Ronald Getoor

RG:ls




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STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94305
DEPARTMENT OF MATHEMATICS

Oct. 26
Dear Ron,

Thanks for the note. The stuff I talked
about in Cq$Jremont would go over the head) of
most people at an IMS meeting. This fact, plus
August Washington D. C. made me decide that I
should decline your invitation. Try another
time when you have the "power". I am certainly
gratified that you voted for me as president.

Protter told that Spitzer said to him: The
Strasbourg school has biult up an elaborate
machinery. Now maybe it is ready to prove that
2 + 2 is 4. This sounds like Frank. He should
be roundly birchffor making such statements
about his betters. A mild guy like you---wS4t
would you say?
If you are keen on it I can organize a ses-
sion for the Vancouver AMS meeting, but I am not
too interested in going there again. It is a
pity that there are so few people who are real-
ly interested in the same things and can learn
from each other. I must exclude Dynkin from
this small sets though he has a lot to learn
from the Strasbourg school indeed.


Sincerely.






STANFORD UNIVERSITY-
STANFORD, CALIFORNIA 94305
DEPARTMENT OF MATHEMATICS


March 20, 1980
Dear Ron,

Your letter about D. is very good. I'll let you know what happens
next. Hans is so stupidly pedantic (or maybe deliberately so in this
case) that he is not aware that deans know better thah us that rules
are made to be bent.
I am annoyed by the blunder in my example. I am so stunned at the
moment I don't even know whether I had a pr6of which covers the case
you mentioned, i. 9. two sigma finite measures on the linear Borel
sets must concede if they agree on compacts. 0'f course the result is
trivial if by sigma finite we mean there is an increasing sequnce of
compacts K (such as [-,.n1) on which both are finite and agree. What
puzzled me at the time I raised the question is whet if we know only
that there are A borel sets on which the measure are finite, and
A increases to space. By the way there is a lot of such junk in the
second part of Halmos but on a casual galnoe it does not seem to set-
tle this question but I may be wrong.
uDurrett gave me his example the day after I cJlled you. If I did
hot tell you (actually I wrote at once) you have'at least gained one
more I think useful insight. For instance I have a neat proof T2.10
on p.93 of Port-Stone using the patixz jart of the tail field result
(terminal times will do). fkamtcorrect Their proof is silly and
misses the intuitive point which is exactly the tail.

Did you hive. the proof that BM killed at ex.it from an open set
is a Hunt process? I am giving the proof in my, MS anyway, if you have
it there I will give a reference. Such a result is too important to
skips that is for BId. Indeed I realized now why it is worthwhile to
generalize to ?!?nt (e-ven your standard) because, precisely that key
example is not Feller when the boundary is not regular. And. of course
one should not assure the bdy to be regular. A.3 a' concrete applica-
tion of uhtt theory it is superb. I am ware of your good result in the
book but I' ar. giving another. proof based on. Doob's treatment: superharmo-
nic postiye = excessive. Cther treatments using approxJmation of su.erh.
functions' by smooth ones are available in Rac's book as well as P/S.
But the ,3upermiartingale approach is in a sense more germane to the sub-
ject.
1By the way there is aseiious commission in the French Note. The
bounday function f in Theorem 1 and other places' is bounded. Doobs
Follmer:, Meyer, Durretts.,... all saw the Iootei without noticing this.
It took' an analyst to ask Mie. 2k-txgaasxfUir It bugged me.

Sincerely*




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JanUary lf, 1988
Ron,
I have just gpt the galley of my MS with ,Iurali, and have done
my part of checking, but will ask him to do his part before sending
it back. If that delays the publication, it does not hurt me.
I am not in the habit of dredging up things, but in going over
the ~:IS ("written 90 per uent or more by myself) I was surprised to note
the inclosed portion (from original MS). In the top few lines I gave
a complete legible proof of the "difficulty" (soi-disant) pointed out
by the referee JI inclose the relevant page for your convenience). In
fact thiA proof covers all powers of A(t), and is by no means news spell-
ed out only in my usual pedagogic (considerate) style. The referee did
not know that easy inequality, which is excusable, but failed also to
read the inclosed portion which contained the general argument. Indeeed
the saie argument is at the back df the so-called Hasminskii inequality
of which ignoranti such as Barry Simon made so mush of. We cannot ex-
pect t e referee to know all about it but it appearsodd that he/she
would wake a fuss over this banality (cf. his/her suggested absolutely
stupid proof in the report). Can we really trust the judgment of a
person who (i) did no, know the elements of the stuff (ii) did not bother
to read (or worse read without comprehending) the MS under review?
Do you not think you owe it to the authors and the referee to tell it
to the latter? Please do so in unmistible terms and let me know the re-
sult. Having been an editor for years I fully realize that such
things occur but when they do they should be addressed. The worst case
in my experience was S. Watanaobe 's reiereeing Ouf cluKean totally wrong
proof published (my fault) in the Z. f. W. Eevm alter iieyer found the
mistake and wrdte Henry the latter did not admit it for six ..onths or ffy
longer. It was I who forced Henry to make an acknowledgment.

Best regards,

rould not this kiad of refereeing give aurali cause of complaint
'for r_ also a proof to me
not really in order to referee that paper full of add. func.


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Referee's report on the paper
"General Gauge Theorem For Multiplicative Functionals"
by K.L. Chung and K.M. Rao

In a previous paper [8] the authors proved the gauge theorem for Brownian
motion. In this paper they have extended the gauge theorem to a fairly broad
class of Markov processes and they have also proved some new results related
to and extending the gauge theorem for these processes. I found Theorems 4, 5,
and 7 to be particularly interesting. I feel that this is a good paper and I
recommend that it be published. On the whole, it is a carefully written paper.
I do have some corrections, comments, and questions, which I will now list.

Page 4, line 1. Delete EX .
Page 4, line 8. Delete sup .
Page 5, line 8. Insert a o between As and et .

Page 7, proof of corollary. I don't believe that E (A ) can be estimated above
in terms of EX[At) (The obvious inequality goes the other way.) However, this
difficulty can be circumvented by means of the inequality aea < a 1/2e /2


7 Page 12, line 12. Why does it follow that D is connected? (The process need
not have continuous paths.)

I Page 12, line -3. There is no remark after Lemma 3. Please clarify. i -. L b
C Page 13, lines 15 to 18. I feel that the continuity assertion requires a more
detailed justification.

Page 15, line -3. Add a d at the end of provide .

Page 16, line 4. (c) is not quite proved in the indicated corollary, since A
/ is assumed to be increasing in that corollary. (The inequality le 11 < ei '- 1
is needed to close this small gap. Perhaps it would be appropriate to mention this)

P./Z,.~Page 16, line 10. The function g has been shown to be lower semicontinuous but
has not been shown to be continuous. Therefore this equality is not obvious. It
& does follow from (26) and the subsequent argument. The reasons given in lines 11
and 12 appear to be inadequate and should be deleted.

Page 17, line 8. Add by Lemma 3 of [3] .

A^
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N/Page
Page
Page
Page
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A Page
7 Page
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type


19, line 13. Change = to < .
19, line -5. Change to and add at the end of the line.
19, line -4. Add a at the end of the line.
23, line -4. Change QD to D and change < 1 to < c < 1 .
23, line -2. Add a } at the end of the line. W 7&cA
28, line 7. Add a ) after [10] .
29, line 4. Change the first the to a
29, line 8. Not quite a general Radon measure. It must satisfy a Kato class
of condition.


/ Page 29, line 9. Spelling error.
v Page 29, line 14. Insert a after have and add a at the end of the line.
V Page 29, line 16. Move (by Proposition 8) to follow at once
oho 'Page 30, line 2. Why should p be even. I think it's enough that it be an integer.L
\ Page 30, line 3. Refer to the proof of Lemma 3.
Page 30, last paragraph. I find that it is stretching a point to compare (46) with
Zhao's result. Surely what is most interesting about Zhao's result is the inform-
ation it gives about how fast 0(x) 0 as x bD (46) gives no such information.


L1. i




STANFORD UNIVERSITY
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Dear Rons
Thanks for the quick and full response. I read Hunt again after you
alerted me, but decided not to say more than in the inclosres for your
edification". I spent a lot of time over ancient literature: Bacheliers
Wieners Levys ... but first of all George Green. My hook is finished!
There is still time/space to add more notes* and correspondence with you
reminded me of the almost forgotten "Condenser Problem". When I gave a
talk o2 / S-k-8s-ki commented mor less: of course he knew it but our
problem was harder/somewhat. Do you understand him? Unfortunatley we
replied on duality to get the results there* unlike my purely last-exit
approach to the equilibrium potential. You and Sharpe [also vaguely Amema]
didnDt connect it with Newton* therefore ti*sing the good part---for me.
However if there is an easy way to extend my doing to the condenser cases
I may still mention it. If you know how, send it quickly. Mypoor Ph. D.
student [in our paper] did some very weak thing not worth a note.
I am inclsoing a discarded (revised) page from my MS for fun. How
do you like my support of Hunt? etc.
If yau come in June. let me know in advance.