Markov Processes: Ray Processes and Right Processes

123 Pages·1975·2.38 MB·English

by  Prof. Ronald K. Getoor (auth.)

Lecture Notes ni
Mathematics
Edited yb .A Dold and .B Eckmann
440
Ronald .K Getoor
Markov Processes:
thgiR
Ray Processes and Processes
IIIII I
galreV-regnirpS
Berlin-Heidelberg • New York 1975
Prof. Ronald .K Getoor
Department of Mathematics
University of California
San Diego
P.O. Box 109
La Jolla, CA 92037/USA
Library of Congress Cataloging in Publication Data
Oetoor, Ronald Kay~ 1929-
Markov processes~ ray processes and right processes.
(Lecture notes in mathematics : 440)
Bibliography: p.
Includes indexes.
i. Markov processes. .I Title. II. Series:
Lecture notes in mathematics (Berlin) ; 440.
qA3.L28 no. 440 [QA274.7] 510~.8s [519.2'33i
75~6610
AMS Subject Classifications (1970): 60JXX, 60J25, 60J40, 60J45,
60J50
ISBN 3-540-07140-7 Springer-Verlag Berlin • Heidelberg- New York
ISBN 0-387-07140-7 Springer-Verlag New York • Heidelberg • Berlin
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PREFACE
The purpose of these lectures si to develop the basic properties fo Ray
processes and their applications to processes satisfying the "hypotheses droites"
of Meyer following the fundamental paper [16] by Meyer and Walsh. Sections 2
through of 7 these lectures discuss the basic results on Ray processes and, in
outline, follow quite closely the presentation in Section of [16]. i However, we
go into more detail than in [16] and, ni particular, we give complete proofs of
the facts needed about resolvents and semigroups in Sections Z and .3
Beginning in Section 9 we give the basic applications fo Ray processes ot
"right processes" again following [16] outline. in However, we depart from
Meyer and Walsh in two important matters. Firstly we assume only that the
state space ~ si a U-space, that ,si a universally measurable subspace a of
compact metric space; whereas Meyer and Walsh assume that E si Lusinien,
that ,si a Borel subspace a of compact metric space. Secondly we do not assume
that the excessive functions are nearly Borel. We assume only that they are
right continuous along the trajectories of the process. This change in the
"hypotheses droites" sometimes requires a modification ni the proofs of the
basic results. Thus the statements fo the theorems in Sections ]0 through 13 are
the same as in Meyer and Walsh, hut often the proofs are somewhat different.
The basic definitions and elementary properties right of processes are given in
Section .9 The Ray-Knight compactification si presented ni Section ,0i while in
Section ii ti si shown that the results on Ray processes developed in Sections 5
through 7 actually hold when properly interpreted for right processes. This si
the most important part fo these lectures. Section 12 contains Shih:s theorem
which was the catalyst for the renewed interest in the Ray-Knight construction.
tI also contains the pleasing result that the excessive functions are nearly Borel
after all, but in the Ray topology. Section 13 discusses the relationships among
right processes, Hunt processes, and standard processes. Finally ni Sections
14 and 15 we investigate ot what extent the preceding constructions are unique.
VI
These results are taken from my joint paper with M. .J Sharpe [6].
The reader of these lectures should be familiar with the general theory
of processes as set forth in the recent book of Dellaeherie [3]. He should also
have some acquaintance with the strong Markov property and the construction of
Markov processes from transition functions as presented in Sections 8 and of 9
Chapter I fo iF~lumenthal and Getoor [l], or in Chapters XII and XIII fo Meyer [9].
However, an extensive knowledge of Markov processes si not required.
We have made no attempt to assign credits for the results discussed here.
Suffice ti to say that lla the of basic ideas come from Knight [7], Meyer [9],
Ray [14], Shih [15], and Meyer and Walsh [16]. Our only contribution si the
relaxation the of assumptions on the state space and the excessive functions and
the results in Sections 14 and 15 as mentioned before. See Meyer []Z] ni this
connection also.
lwould like ot thank M. .J Sharpe for many helpful discussions and sug-
gestions during the writing of Sections 9 through ,31 and ot reiterate that Sec-
tions 14 and 51 are based on our joint paper [6]. C. Gzyl and P. Protter read
most of the manuscript and made innumerable suggestions for improving the
exposition. C. Gzyl also helped with the proofreading of the final typescript.
L. Smith and A. \%'hiteman typed the preliminary and final versions respectively.
Their superb skill greatly eased my work in preparing the manuscript. Fina]ly
I would like ot thank the National Science Foundation for financial support during
part fo the writing under NSF Grant GP-41707X.
R. K. Getoor
La Jolla, California
November, 1974
CONTENTS
I. PRELIMINARIES ................................ 1
Z. RESOLVENTS .................................. 3
.3 RAY RESOLVENTS AND SEMIGROUPS ................. 8
4. INCREASING SEQUENCES OF SUPERMARTINGALES ........ 17
5. PROCESSES .................................. 19
6. PROCESSES CONTINUED ........................... 28
7. CHARACTERIZATION OF PREVISIBLE STOPPING TIMES ..... 33
8. SOME TOPOLOGY AND MEASURE THEORY ............. 43
9. RIGHT PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
10. THE RAY KNIGHT COMPACTIFICATION . . . . . . . . . . . . . . . 57
11. COMPARISON OF PROCESSES . . . . . . . . . . . . . . . . . . . . . . 66
12. RIGHT PROCESSES CONTINUED: SHIH'S THEOREM . . . . . . . . 78
] 3. COMPARISON OF (Xt_) AND (Xt_ ) .................. 88
14. U-SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
15. THE RAY SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Z
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
INDEX OF NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
1. PRELIMINARIES
The reader of these lectures is assumed to be familiar with the general
theory of processes as presented in the recent book of Dellacherie I3]. We
shall refer constantly to ]3[ and adopt the following convention: A reference to
D-If-19 will refer to item 19 of Chapter II in Dellacherie [3]. In addition, the
reader is assumed to have some acquaintance with the strong N{arkov property
and the construction of Markov processes from transition functions as set forth,
for example, in Sections 8 and 9 of Chapter I of Blumenthal and Getoor [i]. How-
ever, an extensive knowledge of Markov processes is not assumed. A reference
to BG-III-(4. 19) will refer to item .4( 19) of Chapter III in Blumenthal and
Getoor [1].
In general, our notation will be the same as that in Btumentha[ and
Getoor. In particular it is assumed that the reader is familiar with the notation
established in Section 1 of Chapter 0 in BG. For example, fi (E,E) is a meas-
urable space and f a numerical function on E, we write E f _E to indicate that
f si E measurable. We let hE denote the bounded real valued E measurable
+
functions on E and b~ (or sometimes bE+ ) the positive functions in b_E.
+
Similarly 6 ~ f (or 6 f E=+) means that f is a positive =E measurable func-
tion on .~] In accordance with modern usage positive means nonnegative. A
numerical function f on ~] is strictly positive fi f(x)> 0 for all x in E. We
tel _E denote the Cr-algebra of universally measurable sets over E_). (E, By a
measure on a measurable space (E, )E_ we shall always mean a positive measure.
fI (E,~I and (F,F> are measurable spaces, then we write ~ f EIF_ _ or
say that f is E IF measurable whenever f is a measurable mapping from
(E,E) to (F,F); that ,si :f E-> F and f-l(B) E E for all B E F. fI ~ a is
measure on (E,E) and f£ E_IF we write ~ = f(~t) for the measure ~ defined
on (F,_F) by v(B) = ~[f-l(B)] for all B 6 F. The measure v = f(~) is called
the image of tL under .f nI( BG this measure was denoted by ~f-i rather than
the more standard f(~).)
If E is a topological space, then the Borel subsets of E are the ele-
ments of the smallest G-algebra containing all of the open subsets of E. We
shall denote this (~-algebra by E or, sometimes, by B(E). The universally
measurable subsets of a topological space are the elements of E = B (E) where
E = B(E) is the (~-algebra of Borel subsets of E. We let ~R denote the real
+ ++
numbers, R~ : [0, o~ ) the nonnegative reals, and R[ : ,0( )oc the strictly posi-
+ ++
tire reals. Then ,_~ R[ , and ~[ are the o-algebras of Bore[ subsets of ,RJ
+ ++
%1[ , and NI respectively.
fI (E,E) and (F,F)_ are measurable spaces, then a kernel K from
(F,F) to (E,E) si a positive function K(x,A) defined for x E F and A E E
such that x-~ K(x,A) si F measurable for each A E E and A-~ K(x,A) si a
measure on (E,_E) for each x EF. The kernel K isfinite fi K(x,E) < oc for
all x and bounded fi sup [K(x, E): x E F} < ~ fI K(x,E) : I for all x EF ,
then K si a lViarkov kernel; fi K(x,E) g i for all x E F, then <I si a sub-
Markov kernel. fI K si a bounded kernel from (F,F) to (E,_E), then
.i( )i *-f Kf; where Kf(x) : K(x,f) = / K(x, dy) f(y)
defines a bounded, linear, positive map from bE to bF such that
+
(1.2,) (f) a bE ; 0 ~ f I f E bE ==>Kf ~ Kf .
n : n
n =
Conversely any bounded, linear, positive map from bE to bF satisfying )-2.1(
is given by a bounded kernel K from (F,F) to (E,E) as in .)i .I(
fI (E,E), (F,F) and (G,G) are measurable spaces and (l a is kernel
from (F,_F) to (E,E) and g is akernel from (G,G) to (F,F), then the com-
position of K and L, LE is a kernel from )__G,G( to (E,__E) defined by
(1.3) LZ(x,A) = / L(x, dy) K(y,A)
for x E G and A 6 E where the integration in (1.3) is over F. If I< and L
are bounded (resp. Markov, sub-Markov), then LK is bounded (resp. Markov,
sub-Markov). By a kernel on (E, E) we shall mean a kernel from (E, E) to
(E, E).
Z. RESOLVENTS
Throughout this section (E,E_) is a fixed measurable space.
.Z( )i DEFINITION. A family (uC~)cc> 0 of kernels on (E, )E_ si a (sub-
Markov) resolvent provided:
)i( ctUC~l ~ 1 for C~> .0
)ii( UC~- U~ = 8( -cL) uCtu ~ for c~,~ > .0
tI si Markov fi c~uC~I = 1 for ¢t> .0
In general we shall omit the qualifying phrase "sub-Markov"; that si a
-i
resolvent will always mean a sub-Markov resolvent. Since ua'(x, E) ~ tc for
all x, each U ~c si a bounded kernel on ,E( E) and so there is no difficulty with
the subtraction in .)ii( The relationship )ii( si called the resolvent equation.
Note that ti si C~U ~c that si a sub-Markov kernel and not U ~c itself.
(Z.Z) RENIARI4S. )a( Itis immediate from )ii( that UO~U ~ = U~U ~ for
,tC ~ > ,0 and that ~ +- U ,x(tc ) • si decreasing and continuous on ,0( ~ .) Con-
sequently U(x, ) = . lira UCt(x, ) • defines a kernel on (E,E_), but, in general,
ct-~0
U(x, ) • need not be finite (or even ~-finite).
)b( fI ~ > 0 andwe define VC~= U ct+$ for ~> ,0 then ti si immediate
that (vC~)ct> 0 si a resolvent and that V = lira Vct = ~ U si a bounded kernel.
*-~C 0
+
)c( fI f 6 E and ~ > ,tc then the resolvent equation implies that
UC~f = U~f + ~( - )~C U c~USf, even though UC~f - USf si undefined in general.
+
)d( tI si immediate from )c( that fi E f E_ , *-~c UCtf si decreasing on
(0, ~ ).
)e( fI f E bE, then frol'n the resolvent equation and .Z( l-i)
Iu~f(x)-u~f(x)l~ I s-~t llflL
aB
where f]l I[ -- sup[ If(x) :I x ~ E} . Consequently ~-~0 U ~C )x(f si continuous uni-
formly in x on each interval ' [C~0 oo ,) I( > 0 .0
)f( fI (UC~)(I>0 si a resolvent on (]E,E), then ti si easy ot check that
(uC~)c~>0 si also a resolvent on (E,E*). This amounts to checking that
x-~ U ~C ,x( A) si $_~ measurable whenever A 6 .'-I~"
+
(Z.3) DEFINITION. t___eL 6 E f and C~- > .0 Then f is c~-supermedian pro-
vided ~U(l+~f ~ f for all ~> .0 ,fI in addition, as ~ >- oc , ~U ~c +~f-+ f point-
wise, then f si 0~-excessive. We tel I(~ (resp. 6 )Z( denote the set of all (l-
supermedian (resp. c~-excessive) functions. We write O S : ~ and = g gO , and
say that f ss_i supermedian .pp_ser( excessive} rather than 0-supermedian (resp.
0-excessive).
The next proposition collects a number of elementary properties of
supermedian and excessive functions.
.Z( 4) PROPOSITION. )i( CC~ and ~cg are convex cones ; G S si closed under
pointwise infima, i.e., fi f,g E zc$ then A f g rnin(f,g) = E ~Cg E I ; .~Cg
)ii( fI E f ~c~ then ~-*~UC~+~f(x) ss_i increasing for each x.
)iii( fI nif ) si an increasing sequence in ~ (rest. ,)~cg then = lira f f
-- -- n
si in 8 tc (resp. .)~cg
)vi( E f tc$ fi and onlyif ~ $ E f for all ~>c~. (The corresponding state-
ment for ~cg si also true - see (Z.8).)
)v( fI C ~ E f then uC~f E ~cg
-- =
PROOF. )i( This is elementary and left to the reader.
(ii) Let f E g~ and fix 0<~ < y. Then ~UC~ ~f ~ .f Applying
(y- ~)U ~C +Y to this inequality gives
~(y_~)U~+YUC~+~f< (¥_~)UCZ+Yf ,
and adding ~U ~c f Y + to both sides find
we
(2.5) $[Ua+Yf+(7-~)uC~+¥UC~+~f] ~ ~UC~+Tf+(y_~)UCt +?f :: yUCt +Yf .
This last equality si clear at each x such that U ~c +7f(x) < oc , but ti si also
clear fi U ~c +Yf(x) = ~ since y> ~ . By the resolvent equation, or more pre-
cisely (Z,Z-c), the left side of (Z.5) reduces ot ~U°~+~f, proving .)ii(
)iii( Suppose ~ 0 f ~ .f fI (f) c g¢~, then ~uC~+~f ~ f . Letting
n n n n
n-> oc the monotone convergence theorem gives ~UC~+~ f ~ ,f and so ~ f gC~
fI each f ~ £~ c ~<~ then )ii( and the definition of ~cg imply that 8uC~+6f f ~
n ' n n
as 6 ~- o= Therefore 6U~+df increases both with n and 6 , and so
n
lim~U(~+Sf : lira lim ~U~+Sf ~ lira lim 8U~+~f ~f.
~ n n n n
Hence £ f .~cg
)vi( in view of )i( and ,)iii( E f ~c~ fi and only f A n E ~Cg for all n -> .I
Consequently in proving (iv) we may assume f bounded. fI 6> ~ and E f ~Cg ,
then for > y ,0 f -> yU¢l+Yf -> yUd+Yf and so E f ~8. Conversely fi yud+Yf ~ f
for all 6>C~, then letting ~ decrease ot ~c and using (Z.Z-e)yields yuC~+Yf < .f
}v{ By )iii( ti suffices to show U°~f E g~ when E f bE .+ But yUC~iYuC~f
= uC~f - U C~+v f ~ UC~f and llf'>+~Cull ~ {0~+v) -I llfll which approaches
zero as y ~- oo (Here, as before, II " II stands for the sup norm. ) Thus
uC~f E gc~.
(Z.6) PROPOSITION. Let E f gc~ Then : f lim 8U~+~f exists and E f g0~.
6->~o
Also f~ f and U6f = Udf for all 6 > .O The function f si called the (c~-
excessive) regularization of f and. si the largest ct-excessive function dorni-
hated by .f
PROOF. We know that 6 >- 6uC~+~f si increasing and dominated by .f Conse-
quently f exists, f< ,f and E f E .+ fI g ~ C~ and g ~ ,f then 8UC~+dg <
8U(l+df, and letting 6 ~" oo we see that g ~ f. Thus f dominates any -~c
excessive function which is dominated by .f Next we shall show that E f g0~
To this end suppose first that f si bounded. By the monotone convergence
theorem Udf = lira Udy UCL+Yf for lla 8> .0 But U~f-U(l+Yf =
o0>-/%
~C( +y-@)uduC6+Yf, and so
7UdU(~+¥f = Udf _ Ua+Tf + ~( -a)UdU(Z+Yf .
Consequently yu6uC~+Yf~ Udf in sup norm as y-~ ~o Therefore U6f = U6f
for all 8> .0 Thus 8UCt+6f =~Uc~+df t f as ~ 8 oc and so E f 8 .~c
For the general case tel f A f = n E gc~. Then 6U~+6f increases both
n n
with n and 6 • Therefore