Generating sweep-out set algebras for Markov processes

Generating sweep-out set algebras for Markov processes

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Simons, F. H., & Overdijk, D. A. (1977). Generating sweep-out set algebras for Markov processes. (Memorandum COSOR; Vol. 7717). Technische Hogeschool Eindhoven.

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Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Generating sweep-out set algebras for Markov processes

by

F.H. Simons an D.A. Overdijk Memorandum COSOR 77-17

Eindhoven, september 1977 The Netherlands

Generating sweep-out set algebras for Markov processes

F.H.

Simons and

n.A.

Overdijk

1. lntroduction

It is not difficult to show that for a dissipative Markov process P on a

probability space (X,E,m) the recurrent sets are dense in E. Moreover examples have been given of dissipative measurable transformations and dissipative Markov processes for which there exists an algebra of recurrent and even of sweep-out sets generating E(cf. [5J, [6J).

Recently

H.

Berbee

[IJ

has shown that a Markov process on

«0,1),8 ,A

)~

where A is the Lebesgue measure on the Borel sets

B

of the unit interval

(0,1),

for which there exists a countable sweep-out partition is isomorphic with a process on

«O,I),B,A)

for which every interval is a sweep-out set. Clearly, this means that for Markov processes on

«O,l),B,A )

the existence of a countable sweep-out set partition is equivalent with the existence of an algebra of sweep-out sets generating E.

In [7J, we gave a necessary and sufficient condition for the existence of arbitrary small sweep-out sets for a Markov process P. In this note we shall show that the existence of arbitrary small sweep-out sets is equivalent with the existence of a countable sweep-out set partition, and either of these conditions implies the existence of a generating algebra of sweep-out sets if E is countably generated.

In particular it follows that for a dissipative Markov process on a probability space (X,E,m) where E is countably generated, there exists a generating

algebra of sweep-out sets.

2. Embedded Markov processes and sweep-out sets

We follow the terminology as used e.g. in the book of Foguel [2J. Throughout, (X,L,m) will be a fixed probability space and

P

will be a Markov process on (X,E,m), i.e. a mapping of M+, the space of equivalence classes of m-almost equal non negative extended real valued measurable functions, into itself such that i) P(

L

n-O 0. f ) '" n n ii)PI s 1.

L

naO 0. Pf n n + (0. ~ 0, f EM), n n

Here, as in the sequel, all statements on functions and sets will have to be interpreted modulo m-null sets. For reasons of convenience we shall suppose PI

= ].

For every A € ~ and every k we have

(2. 1)

=

pk+l 1

=

I .

Here IA means mUltiplication by the indicator function lA of the set A, and AI .. X\A. Hence if we define

(2.2)

<XI

QAf

=

L

(PIA,)n PIAf

n=O

then it is easily verified that Q

A again is a Markov process on (X,E,m),

the

embedded proaeee.

Since PIA(x) can be interpreted as the probability that starting in x we enter the set A in one transition, we can interpret QA1(x) as the probability that starting in x, we ever visit the set A. We start with two propositions which in this interpretation are obvious.

Proposition 2.], If A c B then QAt S QBl S 1. If An t A, then

Q

A _n I t QAl if n + ~. Proof. From (2.1) we conclude

from which the first statement easily follows.

Now suppose An t A. Since (PIA' )kl 1.' S de ' b h·

creas1.ng ot 1.n nand k, we obtain

1 - Q ] A n k = 1 im (P I A' ) 1 -k+<XI k

=

lim lim (PIA') 1

=

1 - lim Q A 1, n+<XI k+<XI n n+<'" n

3

-Proposition 2.2. If QAt ~ a >

°

on AI, then QAt - 1 on X. Proof. From (2.2) we conclude for every k

Q

1 ..

A

Hence if k -+ 00

Since a > 0 and 1 - QAt ~ 0, we obtain 1 - QAt a 0 and QAl a 1.

o

A set A E E for which QAt .. 1 on A is said to be a

P-reaurrent set.

If QAl .. 1 on B, then the set A is called a

P-aweep-out

set for B, if B .. X

then we say that A is a P-sweep-out set. The next proposition is one of the reasons why in general we speak about sweep-out sets without mentioning the process.

Proposition 2.3. Let A be a P-sweep-out set, and B c A be a QA-sweep-out set for A. Then B is a P-sweep-out set.

+

Proof. By [7 J letlllla 1 we have for all f E Ai

Q

f

=

B

Hence QBt

=

on A, and we obtain

00

QBt

=

_{QAI B} + QA1B'

L

(QA1B,)n QAIBI n=O

In the next proposition we use the notation L IA for {B n A

I

BEL}.

Proposition 2.4. Let P be a conservative Markov process on (X,E,m), and let

L. ₁ be the a-algebra oflnvariant sets. Then for every

A E

L the conservative part of X with respect to Q

A is the set A, and the class of invariant sets for QA restricted to A is LilA.

Proof. From the relation

+ for all f E

M

([3J, theorem 3) we easily deduce that the conservative part of X with respect

to Q_A is A n C, where C is the conservative part of X with respect to P. Since C ~ X, this proves the first statement. The second statement now

follows as well from this relation since for any B c A the smallest P-invariant Set containing B is {n~lpn IB > O} and the smallest QA-invariant set containing

B is

Proposition 2.5. Let P be a conservative Markov process on (X,E,m) and let E. be the a-algebra of invariant sets. If R E E. and A c R, then for every

1 1

B E

r

'Wi th B n R

=

A we have Q

A .. QB on A.

P roo. slng f U · [7J ,emma, we ave or every 1 4 h f f '" c M+

00

IAQAf

=

IA

I

(PIA,)npIAf

n=O

QO

=

lAIR

L

(PIA,)n PIAf n=O

QO

o

5

-3. Sweep-out set partitions

The existence of arbitrary small sweep-out sets was investigated in [7J. We shall extend the results of that paper by showing that the existence of arbitrary small sweep-out sets is equivalent with the existence of a

countable sweep-out set partition. To this end we first have to reconsider some of the preliminaries in [7J.

Let P be conservative and let E. be the a-algebra of P-invariant sets. For

1

every R € E. with m(R) > 0 the P-sweep-out set number s(R) was defined by 1

s(R)

=

sup {nl there exists a partition of R into n P-sweep-out sets for R}. If no misunderstanding can arise, we shall omit the mentioning of the process P. By [7J, theorem 1 there exists a partition (X1"",X_oo) of X into invariant

sets such that for every invariant set ReX with meR) > 0 we have s(R)

=

n

n

(1 $ n $ ~). First we show that the converse of [7] lemma 5 also holds.

Proposition 3.1. Suppose n <

00,

and let (AI""tAn) be a sweep-out set partition of X such that riA. ~ L.IA. for 1 $ j $ n.

J 1 J

Then X .. X •

n

Proof. Let Y be an invariant set of X with m(Y) > O. Then (AI n Y, ••• ,A_n n Y) is a sweep-out set partition for Y of n elements, hence s(Y) ~ n.

Suppose that there exists a sweep-out set partition (B1, ••• ,B

n+I) of Y. Let Rjk be the smallest invariant set containing Aj n B

k, and let R be an atom of positive measure of the (finite) ring generated by the R

jk• Then both (R n AI, ••• ,R n An) and (R n BI, ••• ,R n B

n+1) are sweep-out set partitions for R. It follows that there exists a triple (j,k,~) with k

1

~ such that meR n Aj n B

k} > 0 and meR n Aj n B£} > O. From k 1 ~ we conclude

Since EIA. '" r.IA., we have

J 1 J A. n Bk n R _"" A. _{n Rjk n R,} J J A. n BR, n R == A. n RjR, n R, J J hence A. n (R jk n RjR, n R) '" ~. J

Since the latter set is invariant and A. is a sweep-out se::, we obtain

On the other hand, we have m(R

jk n R) ~ meR n Aj n Bk) >

°

and

m(Rj~ n R) ~ m(R n Aj n B~) > O. This contradicts the fact that R is an atom of the ring generated by the R

ik• Hence s(R) ~ n, and therefore s(R) == n. Proposition 3.2. If _{X - Xoo '} then there exists a sweep-out set partition

(A₁,A

2) of X such that the QA ₂ -sweep-out set number of A2 is 00.

Proof. Since the sweep-out set number of X is 00, there exists a sweep-out

set partition (B

1,B2) of X. Because of proposition 2.4 the embedded processes Q

B and QB have conservative parts BI and B2 respectively. Hence, by [7J,

1 2

theorem 1, there exist a partition (Y1"",Yoo ) of BI into Q_B -invariant sets

1

such that every Q

B ₁ -invariant subset R of Y with m(R) _n > 0 has QB ₁ -sweep-out

set number n (1 ~ n ~

(0),

and a partition (Zl""'Zoo) of B2 into

Q

_B -invariant

2

sets such that every Q

B ₂ -invariant subset S of Z _m of positive measure has Q

B -sweep-out set number m( 1 ~ m ~

(0).

2

*

Let us denote by

A

the smallest P-invariant set containing the set

A

E

E.

*

*

Suppose that for some n < 00, m < 00 we have m(Y n Z ) > O. Because of

n m

proposition 2.4 the set (Y*

n

Z*)

n

BI is a Q

B -invariant subset of positive

n m 1

measure of Y

n' Hence there exists a QBI-sweap-out set partition (EI, ••• ,En)

*

*

I !

of (Y n Z) n BI such that for every E.we have E E. = E. (QB) E ••

n m J J l. 1 J

Since Bl is a sweep-out Moreover by proposition

set, by proposition 2.3 every E. is a sweep-out set. J

2.4 we have

E.(QB )

I

E.

=

E.

I

E. for avery j,

l. 1 J l. J

1 $ j $ n, and therefore E.

I

E.

=

E

I

E ..

l. J J

Similarly we obtain a partion~ 1, ••• ,E )of (y* n Z*) n B2 such that

n+ n+m n m

every E. is a P-sweep-out set J

*

*

the set Y n Z has sweep-out

n m

and E.

I

E.

=

E

I

E .• By proposition 3.1

l. J J

set number n+m, which contradicts X

= Xoo'

*

*

It follows that if n < "", m < ""', we have m(Y n Z ) "" 0,

n m

and therefore Now put A]

=

7

-y:

n B2 c:: Zex> if n < "", u (y* n B i) U (Y: n B2) J S;n<oo n d A u (y* B) (y* n B ) - AI an 2· n n 2 U 00 1 - I' I S;n <00

Since (y~, ... ,y:) is a partition of X into invariant sets, and BI and B2 are sweep-out sets, both Al and A2 are sweep-out sets.

Moreover, by proposition 2.5 we have

*

Qy: n BI

=

Q

A2

=

QBl on Yeo n BJ c:: Y • (X)

Hence the Q

A -sweep-out set number of A2 is 00, 2

Proposition 3.3. I f X

=

X_co' then there exists a countable sweep-out set

partition (A

1,A2, ••• ) of X,

c

Proof We proceed by induction on n, By proposition 3.2 there exists a sweep-out set partition (AI,A

I) of X such that the QA -sweep-out set number of I

Al is "". Now suppose (A

I,A2, ••• ,Ap,Ap) of

that we have constructed a sweep-out set partition X such that the

Then by proposition 3.2 there exists

-Q- -sweep-out set number of A

A P

p a

Q

A -sweep-out set partition

p

is 00

(A_p+₁' A_p+_J) of Ap such that the sw:ep-out set number of Ap+1 with respect to the embedded process of

Q-

on A is 00, However, by [7J, lemma 1, the

A p+l

P

latter process is

Q

_A ,and by proposition 2.3 both Ap+l and Ap+l are p+l

sweep-out sets. Hence (Al, ••• ,Ap+l,Ap+l) is a_sweep-out set partition of X such that the Q

A _p+l -sweep-out set number of A _p+ 1 is co. In this way we obtain a sequence A_I,A

2, ••• of pairwise disjoint sweep-out sets. If this

*

00

sequence is not a partition, we replace Al by Al e Al U (n~l An)', which

by proposition 2.1 also is a sweep-out set. 0

We now are ready to prove the main result of this section. For measurable transformations instead of Markov processes

found in [4J.

Theorem 3.1. Let P be a Markov process on a probability space (X,L,m). Then the following statements are equivalent:

i. there exist arbitrary small sweep-out sets.

ii. there exists a countable sweep-out set partition of X.

Proof. The implication ii.~ i.

is

obvious. In order to prove the reverse implication, we first consider the restriction of P to the conservative

part C. Since there exist arbitrary small sweep-out sets, by [7J the invariant set C must have sweep-out number ~ and therefore by proposition 3.3 there exists a partition (C

I,C2, ••• ) of C into sweep-out sets for C.

On the dissipative part D there exists a partition (D

1,D2, ••• ) such that

I

pn I

D. < 00 on X for every i (cf [2], (2.5». Then from

n=O ~ n

o

$ lim (PI D ••• D ) 1 n+oo IU U k-l

we conclude by formula (2.1) that for every k the set CuD u D u ••• k k+l is a sweep-out set. But then by [7], proof of lemma 3, for every i and k the set Ci u Dk U Dk+Iu ••• is also a sweep-out set.

We now construct a partition E

I,E2, .•• of X in the following way. Put

E}

=

Cj U D_1• Suppose the disjoint sets E1, ••• ,E_p have been constructed

such that for 1 s i s p we have

( I ) E.

=

C. U D ~ ~ n. 1+1 1-u. •• u D , n. 1

(2) _;:"!

n ()

_{E.) ;:"! (l - +)m(E.) for I s j}

J 1 J < i.

Define

A

=

C u D u

n p+I np!l ••• u D n and A

=

C p+l u u k=n +1

""

D k p

Then An t A. Since A is a sweep-out set, we have by proposition 2.1

QA t 1 if n + "", and we can find an integer k such that

n

m({Q~ ;:"! nnE.) ;:: (l - - ) m(E.) for I s J s p.

J p+l J

Put Ep+!

=

~, then the p+l disjoint sets Ej, .•• ,E

p+1 also satisfy (I) and (2). Obviously, (E},E2, ••• ) is a partition of X. Moreover, let A be any set

consisting of infinitely many of the sets E

k, Then from (2) and proposition 2.1 it follows that QAt ~ ~ on every Ej' and therefore by proposition 2.2 the set A is a sweep-out set.

Hence every countable partition of the natural numbers into countable sets

9

-4. Generating algebra's of sweep-out sets

Let P be a Markov process on a probability space (X,t: ,m), and OL an algebra of measurable sets. The algebra is said to be generating if~ is dense in E wi th respect to m, and Cl.. is said to be a sweep-out set algebra if every element of

ot.

of positive measure is a sweep-out set.

If there exist arbitrary small sweep-out sets, then for every n there exists a sweep-out set B with m(B ) <

1..

Then for every A Eo" E by proposi don 2. I

n n n

the set A U Bn is sweep-out set, and therefore the sweep-out sets are dense

in L

We shall show that, if E is countably generated, we can select from the class of sweep-out sets a sweep-out set algebra generating E.

Theorem 4.1. Let P be a Markov process on a probability space (X,E,m) and suppose that E is countably generated. If there exist arbitrary small sweep-out sets, then there exists a generating algebra of sweep-sweep-out sets.

Proof. Since there exist arbitrary small sweep-out sets, by theorem 3.1 there also exists a sweep-out set partition (A

1,A2, ••• ) of X.

For every n the a-ring EIA is countably generated. Since for any generating

n

sequence BI,B_Z"" the sequence X, BI,Bi, B2 n B_I,

Bi

n B_I, B2 n Bi, B2 n Bi,'" is also generating, we may suppose that

riA

is generated by a system of

n

sets ( A . , ) with 1.

= 0

or 1, such that ( A . . O,A. i i

k _ 1 1) nj 1. 1,,·1.k J n;11, •• lk_1 n, ••• is a partition of A " . (k 2 I), n, 1 1 , • .1.k- 1

We now construct a similar system of sweep-out sets of X.

We start by taking a partition (E₁;O' E_1j1) of X such that A1;O C E1;O'

AI;) C E_{I ;1'} and for n 2 2 every An is contained in El;O or El;]

both EI;O and El;l contain infinitely many An' Then (E1;O' EI;l) sweep-out set partition of X.

such that is a

We now proceed inductively. Suppose that after the n-th step of the construction we have obtained a partition (E

k.. ,1 . ), 1 $ k $ n, i.

=

0 or 1 (lsjsn), such that

1, · · · 1 .n J

(i) each E

k;1. . contains infinitely many A. with j > n.

1,··1.n J

(ii) each A. with j > n is contained in one of the sets E . .

J k;1. 1,··1.n n A.. . " E_k · . n J'~l A J .• --k;1 l

···1

n ;1.1, .. 1.n (iii)

(iv) the partition (E

k; 11 ••• 1 . . ) is a refinement of the partition n

(E

k;1.· . ) •

First note that A _{n+1 •} 's _con t ' d' _{a~ne ~n exact y one} 1 ₀ f h E ' _{t e sets k;~l"'~n'} . say Ek,.·, _{. ,l},.,ln} ·.,This set ~s split up into 2n+1+2 disJ'oint subsets • _.

E t " • '0' E E wi th i, :: 0 I f 1 ~ . < + I

k ;ll,·,ln k';ii •• ,i~l' n+l;i

l ,.,in+1 J or or J - n ,

such that condition (i) and (iii) with n replaced by n+l hold for each of these sets, and every A

J. with j > n+l and AJ

• C E , . , '" belongs to one

k ;1

1, •• ln

of these sets. Every other set Ek' . lS split up into two subsets

; 1

1, .. In

Ek •· _,It,,,ln . 0 and R'l' _{K, 1'"} l' _n 1 such that condition (i), (ii) and (iii) hold with n replaced by n+l. Condition (iv) is now automatically fulfilled.

Let the class

ex

consist of finite unions of sets E

k" ,11,,·ln . , together wi th

0.

Obviously the class

at

is closed under the operation of taking unions.

From (iv) it follows that every element of t5L can be written as a finite disjoint union of sets E

kill,·,lm ; . with m fixed, Since the sets Ek . ;ll",lm ,

form a partition of X, the class tlL is also closed under the operation of taking complements, and therefore OL is an algebra.

Because of condition (i) every non-empty element of ~is a sweep-out set, and therefore Ol is a sweep-out algebra,

In order to show that Ol is generating, it suffices to show that every set

A . ,can be approximated by elements of OL,

-K.; ~ 1 ' •• In

Take m 2 max(k,n) and choose £ > O. Put

E

=

u i _n+l -0

u i -0

m

Then E E ct., and by condition (iii)

E n m u j=l A. == J _i u ₌₀ n+l U A., . . 0 --k, 1 1, , ,1 1 == m m

Hence if we choose m so large that mC u

j=m+l

A.) < £, we have meA ~ E) < E.

J

In [7J we have shown that for a dissipative Markov process on (X,~,m) there exist arbitrary small sweep-out sets. Hence if P is a dissipative Markov process on a countably generated probability space (X,E,m), then there exists an algebra of sweep-out sets generating E,

- 11

-5. References

[IJ Berbee, H.: Personal communication,Free University, Amsterdam, 1977.

[2J Foguel, S.R.: The ergodic theory of Markov processes. Van Nostrand

Mathematical Studies

#

21, New York, Van Nostrand Reinhold, 1969. [3J Groenewegen, L.P.J., K.M. van Ree, D.A. Overdijk and F.R. Simons:

Embedded Markov processes and recurrence. Internal publication CaSaR 75-03. Eindhoven University of Technology 1975.

[4J Helmberg, G. and F.R. Simons: Aperiodic transformations.

Z • Wahrscheinlichkeitstheorieverw-=-·G~1;:.T:r~""180";190 "(1969)

, ,.

-[5J Simons, F.R.: Recurrence properties and periodicity for Markov processes.

Ph.D. Thesis, Eindhoven University of Technology, 1971.

[6J Simons, F.R. and D.A. Overdijk: A recurrent dissipative Markov process.

Internal publication CaSaR 77-02. Eindhoven University of Technology, 1977.

[7J Simons, F.R. and D.A. Overdijk: Sweep-out sets for Markov processes.

Internal publication CaSaR 77-04.Eindhoven University of Technology, 1977.