On a class of embedded Markov processes and recurrence

On a class of embedded Markov processes and recurrence

Citation for published version (APA):

Simons, F. H. (1976). On a class of embedded Markov processes and recurrence. (Memorandum COSOR; Vol. 7610). Technische Hogeschool Eindhoven.

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EI~~HOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 76-10

On a class of embedded Markov processes and recurrence

by

F.H. Simons

Eindhoven, July 1976 The Netherlands

On a class of embedded Markov processes and recurrence

by

F.R. Simons

Abstract. By means of a general type of embedded process we shall give a short deduction of some recurrence properties of the Markov shift.

1. Preliminaries

Let (X,L,m) be a a-finite measure space. Let M+ be the space of (equivalence classes of almost everywhere equal) nonnegative extended real valued measurable functions on X. A Markov operator is a mapping P of M+ into itself such that PI :s; I, and P(

I

ex f )

=

n=1 n n 00

I

a Pf n=1 n n + f E: H , ex n n ~

o.

The domain of P can be extended to L_oo such that P is a positive linear contraction in L_oo Such an operator is always the adjoint of a positive linear contraction in L1, which we shall also denote by P, but now written to the right of the function symbol. The action of this positive linear contraction P on L1 can,

+

by means of monotone approximation, be extended to the space M • It follows that <fP, g> = <f ,Pg> _{for all f, g E:}

_M

+ •

Here <f,g> stands for !fgdm.

With respect to a given Markov operator P on (X,!:,m) we can decompose the

space X into a conservative par.t C and a dissipative part D. This decomposition is __e to E. Hopf, and can e.g. be found in [1], chapter II, [5],

chap-ter 4, § 2. For later use we collect the results which we shall need in two

lemma's and some corollaries.

Lemma 1. The following statements are equivalent.

i. The conservative part of X with respect to P is C.

ii. The set C is the (mod m) largest set such that for all subsets A we have

on A.

2

-Lemma 2. The following statements are equivalent.

1. The dissipative part of X with respect to P is D.

ii. The set D is the (mod m) largest set such that there exists a function g ~ 0, with {g > O} = D and I:=opng is bounded.

(This equivalence can be obtained e.g. from (2,5) in [1] and the maximum principle, chapter 2, theorem 1.12 in [5J).

Corollaries. I • 2. PI = PI = I on C, and therefore PI D = o on C. C 00 +

_L

_pnf

For every f E M we have = 0 or 00 on C. n=1

00 co

I f

_Co

= {

I

pnf = O} n C, and C_I = {

I

pnf = oo} _{n C, then PIC.} =

o

on

n"'l n=l 1

C

I-1., i=O,I.

3. If A c C and m(A) > 0, then < Ic,PI

A> > O. It follows that {lcP > O} = c. 4. For every s ~ I the conservative part of X with respect to pS equals C.

2. The embedded processes

s-I

Let P, Hand H' be Markov operators, and assume H + H' • P for some s ~ I. We define the operator Q by

00

Q =

L

(PH')npH n=O

Obviously Q is a a-additive mapping of M+ onto itself. The next lemma implies by substituting f = ) that Q is a Markov operator.

+

L~maa 3. If for f E

M

we have Pi $ i, then we also have Qf $ f.

Proof. Using pSf $ f, we easily verify by writing out

n-I

I

(PH,)rpHf + _(PH,)nf $ f.

r=O

3

-If H is the multiplication by the characteristic function of a set A~ H' the multiplication by lA" then Q is the embedded process on the set A. The situation that H is multiplication by a function f, 0 ~ f

s

1, and H'

multiplication by the function 1 - f. is studied in [2J, [4J.

o

In both cases we have H + H' = P , hence s = I. The situation with s > I

occurs when we are investigating recurrence properties of the Markov shift, as We shall see in the next section.

Theorem 1. For every f E

M

+ we have

00 00

I

Qnf

₌

I

pns+lHf. n=1 n=O Proof. p(n-l)s+I Hf

=

P(H' + H)P P(H' + H)PHf n n =

L

_{(PH') IpH{PH ' ) 2}pH nl+···+nk+k=n and therefore 00 n =

L

L

(PH') IpH n=l nO+ ••• +nk+k=n n (PH') k pHf n (PH') k pHf

..

00 00 00 n ~

L

(PH I) IpH ••• (PH') "'pHf ~=O

Theorem 2. The conservative part of X with respect to Q is {leH > O} n C, where C is the conservative part of X wi th respect to P.

4

-Proof.

L Suppose Ac {I H > o} n C. c

By theorem 1 we have

It follows by corollary 4 and corollary 2 that this sum is 0 or ~ on C. Put 00 Co

= {

I

Qn_1A

=

o} n C, n=l 00 C 1

= {

I

Qn1A = oo} n C, • n=l then pSI

Co = 0 on C1' and since PHI A

_o

we obtain

00 ~

I

pns pH1A

=

_{o on Co} n=O ~

I

_{pns pH1A} n=O 0 co ~

I

pnspsl

c

= 0 on CI

,

n=O 0

hence in particular PHI

A

_o

= 0 on C.

This means <lCPH1A >

=

0 , and therefore by corollary 3

o

S~nce A

Oc A, we have m(Ao) = 0, and

on A.

Hence by lemma 1 the set {lCH > O} n C is a subset of the conservative part of X with respect to Q.

1.1.. 1£ A

that

- 5

-O} n C, then <ICHI

A> = 0, hence RIA = 0 on C. It follows

o

on C,

hence A is a subset of the dissipative part of X with respect to

Q.

iii. Let g be a function with {g > O} == D and

r

_n=OPn_g is bounded. Then

0:> 00 00 00

L

Qng =

L

pnspHg ~

L

pnsg ::;

L

png < 00

,

n=1 n=O n=1 n=O

and D is a subset of the dissipative part of Xwith respect to

Q.

Corollary. If s

=

1 and H is multiplication by a function h with 0 ~ h ~ I,

then {lcH > O} n C

=

{h > O} n C and we obtain a result of Lin [4].

3. Recurrence properties for the Markov shift

In this section we shall give, with the aid of the operator

Q

of the previous section, a fast deduction of some recurrence properties of the Markov shift. These results go back to a paper of Harris and Robbins [3J.

Let S be a measurable transformation on a measure space (n,F,M). A set A is

said to be uandering under S if no points of A return to Aunder the action of S,

and recurrent if M-almost all points of A return to A under the action of S.

A set is said to be dissipative if it is (mod M) the union of countablymany wandering sets, and conservative if every subset is recurrent.

Obviously, if A_n is the subset of A of the points which return exactly n times under S to A then A is wandering. Hence, if almost all points of Areturn

n

finite ,. many times to A, then A is dissipative.

Now let (~,F,M) be the realization space of Markov proces P on

(x,I>

with initial probability m, i.e. (n,F)

n:=o(X,L),

and

... PIA I>

6

-the i-th coordinate. X-IE and let S be the

n '

for all AO, ••• ,A E E, where X. denotes projection on

n 1 -I

Let F_n denote the a-algebra generated byX

_o

L: , ••• ,

shift transformation in (n,F).

Suppose that the initial measure is such that F can also be con'sidered as a

+

Markov operator on M (X,E,m). (This is the case if and only if meA)

=

0 ~ - P(·,A) = 0 m-a.e.) Let C be the conservative part of X with respect to P,

00

and define C = {X ~ C for all n}.

n

Theorem 3.

00

i) The set n\C is dissipative.

ii) For every n and every A E

F ,

the set A

n

COO is recurrent.

n

Proof. Because of Pin = 0 on C, we have

00

n\Coo = {X

OE D} = u {XOE Di } ,

i=1 where D

I,D2,. .. is a partition of D such that

tence of such a partition easily follows from

00 00

L

M(X E D. ) :=

I

_<IPn_I D.> < 00

,

n 1 n=O n=O 1. 00

I

pnl is bounded. The exis~

O D.

n= 1.

lemma 2. We then have

hence by the Borel-Cantelli lemma we obtain M{X E D. i.o.}

=

O. Almost all

n 1.

points of {X

OE Di} return to this set under S only finitely many times, so

00

{X

_o

E D

i} is dissipative, and therefore {XOE D}

=

u {XOE Di} is also dis-i=1

dipative.

00

ii) Without loss of generality we may assume that X = C, and therefore C n. Ch_{00 ~} A _E F I' and d f ·e lne for every f c_~ M+

s-where we consider the FO~easurablefunctions on nin the right-hand side as functions on X.

The operators Hand H' are Markov operators on (X,E,m) and satisfy (H + H')f = E(f(X

7

-Let A

Obe the set of points of A which return to A under SS at least once. Then u n=l ns } i ~ n, S W E: A <Xl M(A O) =

l

<IH(PH,)n-l pH1 > = <IHQ1> • n=1

since by theorem 2 Q ~s conservative on {IH >

a},

we have QI

=

I on {IH > OJ,

and therefore M(A

O)

=

<IH1>

=

M(A) < <Xl, hence AO

=

A (mod M). Hence the set

. s

A ~s recurrent under S , and therefore under S. Remark I. Let P be conservative. If A E F

s' then A E: Ft for every t

z

s, and we have actually shown that A is recurrent under st for every t. Hence almost all points of A return to A infinitely many times underS.

Remark 2. The crucial point in the paper of Harris and .Robbins is that if there exists an algebra of recurrent sets generating

F

and a finite or a-finite equi-valent invariant measure, then

n

must be conservative. Therefore, if there exists a function u with 0 < u < <Xl on C, u

=

0 on D and uP

=

u~ then the

mea-dM'

sure M' defined by ---

=

u(X

O) is (o-)finite and invariant under S, and the

dM

00

set C is conservative.

References

[IJ Foguel, S.R.: The ergodic theory of Markov processes. New York: Van

Nos-tran~ Reinhold Company, 1969.

[2J Foguel, S.R., Lin, M.: Some ratio limit theorems for Markov operators. Z. Wahrsch. verw. Geb. ~, 55-66 (1972).

[3J Harris, F.E., Robbins, H.: Ergodic theory of Markov chains admitting an infinite invariant measure. Proc. Nat. Acad. Sci. U.S.A. ~, 860--864 (1953).

[4J Lin, M.: On quasi-compact Markov operators. The Annals of Probability

l,

464-475 (1974).

[5J Revuz, D.: Markov Chains, Amsterdam: North-Holland Publishing Company, 1975.