The recurrence behaviour of random walks on locally compact groups

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Overdijk, D. A. (1979). The recurrence behaviour of random walks on locally compact groups. (Memorandum COSOR; Vol. 7906). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

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PROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCH, AND SYSTEMS THEORY GROUP

Memorandum COSOR 79-06

The recurrence behaviour of random walks on locally compact groups

by

D.A. Overdijk

Eindhoven, May 1979 The Netherlands

D.A. Overdijk

Introduction and summary

+

Let (x,I) be a measurable space and let

M

be the class of nonnegative ex-tended real valued measurable functions on (X, I). A Markov process on (X,~) is a mapping P of M+ into itself such that

( 1) 00

I

n=i a. pf n n P 00

I

n=i a. f n n (a.n + ~ 0, f EM) , n ( 2) Pi ~ 1

If we put P(x,A) PiA(x) for every x E X and every A E I, then P(·,·) is

the (sub) transition probability describing the process on the state space

(X,I). Conversely, every (sub) transition probability on (X,~) determines

a Markov process on (X,I).

Sometimes we want to consider Markov processes in a more global sense i.e. modulo a a-finite measure ~ on the state space (X,I). Let (X,I,~) be a

a-+

finite measure space and let

M

(~) be the space of equivalence classes of

~-almost everywhere equal nonnegative extended real valued measurable func-tions on (X,I). A Markov process on (X,I,~) is a mapping P of M+(~) into itself such that (1) and (2) hold ~-a.e.

When P is a Markov process on a measurable space (X,I) and ~ is a a-finite measure on (X, I) , then in general P is not a Markov process on the measure space (X,I,~). This is the case if and only if P is nonsingular with respect to ~ i.e. for every A E I with meA) = 0 we have m({Pi

A >

oJ)

= O. Conversely, if P is a Markov process on a measure space (X,L,~), then in general there does not exist a (sub) transition probability on the measurable space (X,I) describing the process.

In this note we shall discuss the recurrence behaviour of a random walk on a locally compact abelian group (G,I), where ~ denotes the a-algebra of the Borel sets of G.

Our main tool to investigate the recurrence behaviour of random walks will be the use of embedded processes. Let P be a Markov process on a measurable space (X,~), then for every A E L the embedded proces P

A is the Markov pro-cess on (X,I) given by

00

I

n=O

+

Here AI stands for x\A and I for multiplication by the indicator function

A

1 of the set A. The expression P 1 (x) is the probability that, starting

A A B

in x, at its first visit to the set A the process will enter the set B. Section one gives preliminaries concerning Markov processes. In section two we shall show that a random walk is either conservative or dissipative and

the random walk is conservative if and only if the random walk is recurrent. Section three gives criteria for a random walk to be Harris (cf. [5J, ch. 3 theorem 4.4) and to be ergodic. In section four some recurrence criteria for random walks onmn (cf. [2J) will be given.

1. Preliminaties concerning Markov processes

In this section we collect some results about Markov processes, which can be found in [7J. P will be a Markov process on an arbitrary measurable space

(X,I). A a-finite measure ~ on (X,I) is said to be invariant with respect to

P if

J

Pf(x)~(dx)

J

f (x)

~

(dx) for all f

EM.

+

Proposition 1.1. If P is a Markov process on the measurable space (X,I) and the a-finite measure ~ on (X,I) is invariant with respect to P, then P is nonsingular with respect to ~ and therefore P can also be considered as a Markov process on the measure space (X,L,~).

Proof. Suppose A E I with ~(A)

=

O. Since ~ is invariant we have

Hence PiA

o

~-a.e. and therefore ~({P1 >

a})

A O.

o

Proposition 1.2. If P is a Markov process on the measurable space (X,I), then for all sets A c I and all f E M+ we have

y

n=l

Proof. Replac~ in [7J proposition 2.2 A by AI, B by A and f by IAf.

0

Let ~ be a a-finite measure on (X,I) such that the Markov process P on (X,I) is nonsingular with respect to ~, then P is also a Markov process on the mea-sure space (X,I,~). The decomposition theorem of E. Hopf states, t~at there

exists a mod ~ unique decomposition of the state space X in a conservative part C and a dissipative part D ([3J, ch. II). In the following theorem we collect some important properties of the conservative and dissipative parts.

Theorem 1.1. Let P be a Markov process on the a-finite measure space (X,L,~) and let C and D be tile conservative and dissipative parts of X with respect to P, then we have

i) For all A E L with Ace

!

pnl

A

=

00

~-a.e.

on A . n=O

ii) There exists a measurable partition of D ql ,q2' ... such that PD. 1 ~ qi < 1 ~-a.e. on D_i ~ D 1 u D2 u ••• and a sequence 00

I

n=O 1

~ ~-a.e. on X for all i • 1 - q.

~

For a proof of theorem 1.1 see [7J, theorem 2.1 and 2.2.

Theorem 1.2. If P is a conservative Markov process on a measure space (X,L,~)

+

and for a function f E

M

we have Pf ~ f ~-a.e., then Pf

=

f ~-a.e.

For a proof of theorem 1.2 see [3J, ch. II theorem B.

Let P be a Markov process on a measure space (X,L,~). A set R E L is said to

be invariant, if P1

R 2 lR ~-a.e. Intuitively, this means that i t is

impossi-ble to leave the set R under the action of the process. There is no unanimity in the definition of invariant set. The definition in [3J differs from our definition, but for conservative Markov processes they are equivalent because of theorem 1. 2.

Proposition 1.3. If P is a conservative Markov process on a measure space

*

(X,L,~), then for every A E L we have PAl

=

l

A

*

~-a.e., where A is the mod ~ smallest invariant set containing A.

Proposition 1.4. Let P be a Markov process on the measure space (X,~,~) and

th th th {dV > O} b 1

v a finite invariant measure wi v«~, en e set d~ e ongs to

the conservative part.

For a proof of proposition 1.4 see [3J Ch. IV theorem E.

If P is a Markov process on a measure space (X,~,~), then i t is easily veri-fied that for every A E ~ the embedded process P is also a Markov process

A

on (X,~,~). Let C(P) be the conservative part of X with respect to P and C(P

A) the conservative part of X with respect to PA'

Proposition 1.5. Let P be a Markov process on the measure space (X,~,~), then for every A E ~ we have C(P

A) = A n C(p) .

For a proof of proposition 1.5 see [7J, proposition 2.4.

Proposition 1.6. Let P be ~ Markov process on the a-finite measure space

(X,I,~) such that ~ is invariant with respect to P. If A E I with 0 < ~(A) <00

and PAl = 1 ~-a.e. on A then A c C(P) •

Proof. The restriction of ~ to the set A will be denoted by ~A' We shall prove that ~A is invariant with respect to P

A" Using the invariance of ~ with respect to P, it is a straightforward verification by induction, that for all N E ~ and all bounded functions f E

M+

we have

Hence

(1)

for all B E I . Define the measure v on (X,~) by

v(B) =

J

IAPA1B(x)~(dx)

Since v «

~

the Radon-Nikodym derivative

~~

exists and from (1) we conclude dv < 1

~-a"e.

d~ - A

Since PAl = 1 ~-a.e. on A we have

d\l

Hence - -

=

1 ~-a.e.

d~ A

+

For every f E M we now have

J

PAf(x)~A(dx)

=

f

=

J

=

f

f (x)~A(dx) •

Hence ~A is invariant withe respect to P

A• From proposition 1.4 we conclude A C C(P

A) and by propositior11.5 we have A C C(P). , 0

2. Recurrence and conservatively for random walks

Throughout the sections two and three G will be a locally compact abelian group. We shall suppose that G is metrizable and d is a metric on G compati-ble with the topology. The group operation in G will be written additively. The a-algebra of the Borel sets of G will be denoted by L and the Haar mea-sure on (G,L) by A.

A random walk on (G,L) is determined by a probability measure p on (G,L). Following Revuz (see [5J, p. 27) the probability measure pis called the law of the random walk. A random walk with law p is a Markov process on

(G,L) with transition probability P(x,A) =

J

1

A(x+Y)P(dy) for all x i G and all A E L •

The corresponding Markov operator P on M+ is

Pf(x) =

f

f(x+y)p(dy) for all x E G and all f E M+ •

In the sequel we shall frequently use the translation operator T (a E G)

a

M

+.

on For every a E G define

T f (x) = f (x+a)

a

Proposition 2.1. Let P be and every A E L we have

i) PT_a

=

T P

,

a

ii) _PATa T P

a A+a

. M+ for all x E G and f E •

a random walk with law p, then for every a € G

+

Proof. For every f E M we have

PT f.(x)_a =

f

T f(x+y)p(dy)_a =

J

f(x+y+a)p(dy) = T Pf (x)

Hence the operators P and T commute, which proves i). For every BEE we d. have Hence We conclude 1 B+a(x+a)f(x+a) T Ia B+af (x) • P T A a 00

L

n=O

o

Proposition 2.2. The Haar measure A is invariant with respect to every ran-dom walk.

+

Proof. Let P be a random walk with law p, then for every f E

M

we have using Fubini 's theorem

f

Pf(x) A(dx)

f

(J

f(x+y)p(dY))A(dx)

=

J

(f

f(x+Y)A(dx))p(dy)

f

f(x)A(dx)

f

p(dy)

= J

f(x)A(dx) •

0

Proposition 2.3. If A,B E E with A(A) > 0 and A(B) > 0, then there exists a set C E L withA(C) > 0 such that A«A+p) n B) > 0 for all p E C.

Proof. Suppose A( (A + p) n B)

=

0 A-a. e. on G. Then we have

o

=

J

.>"«A+p) n B)A(dp) =

f

(f

1_A(x-p)l_B(x)A(dX))A(dp)

= A(-A) A(B) > 0 •

This contradiction proves the proposition.

o

Let P be a random walk on (G,L) with law p. Because of the proposition 1.1 and 2.2 the random walk P is also a Markov p~oces on the measure space

(G,X:,A) •

Theorem~. Let P be a random walk on (G,L) with law p. The random walk as

Markov process on the measure space (G,L,A) is either conservative or dissi-pative.

Proof. Let C and D be the conservative and dissipative parts of C with res-pect to P and suppose A(C) > 0 and A(D) > O. It follows from theorem 1.1 that

there exist a Borel set A c D and a number s such that A(A) > 0 and

I:=o

pn_{l A}

~

s A-a.e.

From proposition 2.3 we conclude, that there exists a point pEG such that A«A+p) n C) > O. Put B = (A+p) n C, then B c

c,

A(B) > 0 and by proposi-tion 2.1 00 00

I

n=O 00 00 T \ pn1 ~ s A-a.e. -p L A n=O Since B c C we conclude from theorem 1.1 A(B)

the theorem.

O. This contradiction proves

o

Remark~ When we say a random walk is conservative (dissipative), t~e random

walk is considered as a Markov process on the measure space (G,L,A).

Let P be a random walk on (G,L). A set A E L is said to be recurrent if for

every starting point x E A the random walk will return to the set A with probability one i.e. PAl (x) = 1 for all x E A. The random walk P is said to be recurrent if every open set is recurrent (cf. [1

J,

def. 3.31). Let fl

be a a-finite measure on (G,L), then a set A E L is said to be fl-recurrent if PAl = 1 fl-a.e. on A. It follows from theorem 1.1 that the random walk is conservative if and only if every set A E L is A-recurrent.

One might think, that a recurrent Markov process is conservative. This, how-ever is not the case, there exists a recurrent Markov process on OR,L,A), which is dissipative (see [6J). For random walks however recurrency and con-servativity are equivalent.

Theorem 2.2. A random walk is conse~vativeif and only if the random walk is recurrent.

Proof. First suppose the random walk is conservative. Let A be an open set such that there exists a point pEA with PAl (pi

=

s < 1. For all E > 0 and all x E G put Ue(x) = {y E G

I

d(x,y) < e}.

Choose e > 0 such that U

2e(p) c A.

Because of proposition 2.1

we

have for all q E U (p) E

P U ( ) 1(q) £ p = P_u (p) _q+pl (p) £ P T 1(p) U (p)-q+p q-p £ s < 1 •

By theorem 1.1 we have U (p) belongs to the dissipative part of G and hence

£

A (U (p» =

o.

£

This contradiction proves that the random walk is recurrent.

Conversely, suppose the random walk is recurrent. Let A be an open set with

a

< A(A) < 00. From proposition 1.6 we conclude that A belongs to the vative part of G. It follows from theorem 2.1, that the random walk is conser-vative.

Corollary. A random walk is conservative if and only if there exists a A-recurrent set A E E with

a

< A(A) < 00.

D

Proof. If the random walk is conservative then it follows from theorem 1.1 that every set A E E is recurrent. Conversely, suppose there exists a

A-recurrent set A E E with

a

< A(A) < 00, then it is an immediate consequence of proposition 1.6 and theorem 2.1 that the random walk is conservative.

D

3. Harris recurrence and ergodicity

We start with some definitions and properties of probability measures on

(G, E) •

If ~ and v are two probabilities on (G,E), then the convolution ~

*

V is a probability on (G,E) defined by

(~

*

v) (A) ==

J

J

l

A(x + y)

~

(dx)v(dy) for all A E E • *n

The n-fold convolution of ~ will be denoted by ~

Proposition 3.1. If ~ is a probability on (G,E) with ~ « A, then for all probabilities V we have ~

*

v « A.

Proof. Suppose A E E with A(A)

=

O. Since ~ « A and A(A)

~(A- y)

=

a

for all y E G. Hence

a

we have

q + r with q « A and r singular with respect to A.

n n n n

Let ~ be a probability on (G,E), then for n ~ 1 there exists the

decomposi-. *n

t~on ~

Proposition 3.2. If ~ is a probability on (G,E), then r ~ r

*

r for all

n+m n m

nand m.

Proof. _~

*

(n+m)

₌

_~*n

*

_~*m

₌

(q +r )

*

(q +r )

n n n m

I t follows from proposition 3.1 that r ~ r

*

r .

n+m n m IJ

Following Revuz ([5J, p. 91) we say that a probability ~ is spread out, if

*p ,

there exists an integer p such that ~ is nonsingular with respect to A

i.e. q_p ~

o.

Proposi tion 3.3. If the probabili ty ~ on (G,E) is spread out, then r (G) .j..

a

n

if n -+ 00.

Proof. By proposition 3.2 we have

r 1 (G) ~ (r

*

r) (G)

n+ n

Hence there exists a number g ~ 0 such that r (G) .j.. g if n -+ 00.

n

Since ~ is spread out there exists an integer p such that r (G) = s < 1. From

p proposition 3.1 we conclude for all k

Hence g

o.

k

s

o

In the sequel we shall use the following well known fact (see e.g. [4J, ch.

3, § 6.1).

Proposition~. Given are two functions f and g on G such that f E £1 (A)

and g E £oo(A). The function ¢(x) =

J

f(x + y)g(y)A(dy) is a continuous

func-tion on G.

A random walk P on (G,E) with law p is called recurrent.inthe sense of

Harris (or shortly Harris) if for every starting point x E G and every A E E

i.e. for every A E E with A(A) > 0 we have PAl (x)

=

1 for all x E G (cf. [5J, ch. 3, de£. 2.8).

The random walk is said to be sp~e~dou~ if the law p is spread out.

Theorem3~1. A random walk P with lawp on a connected group G is Harris if and only if the random walk is recurrent and spread-out.

Proof. First suppose the random walk is Harris.

It follows from theorem 1.1 that the random walk is conservative and by theorem 2.2 recurrent.

Suppose the random walk is not spread out. Then there exists a set A E

E

*n

with A(A)

=

0 and p (A)

=

1 for all n. Then we have

P A,l(O) = 00 00

L

n=O

J

*n .

lA' (y)p (dy) = 0 •

Since A(A') > 0 we conclude that the random walk is not Harris. This contra-diction proves that the random walk is spread but.

Conversely, suppose the random walk is recurrent and spread out. Take AE E wi th A(A) > O.

Since PP_Al

=

PI

APAl + PIA,PAl

=

PAl - PIA(l - PAl) there exists a function

M+ n

g E such that P PAl

+

g if n + 00. Because of the a-additivity of P we have

Pg = P (lim pnpA1) n+oo g • Hence Pg

=

g. We now conclude g (x + y) r (dy) n

f

g(x+y)r n (dy) g(x+y)qn(dy) +

f

dq n g (x + y) dA (y) dy +

=

png (x)

=

f

g (x + y) p*n(dy)

f

J

g(x) Since g ~ 1 we have

J

dq

o

~ g(x) - . g(x+Y)dAn(y)dY ~ rn(G) •

J

dq

g(x) = lim g(x+Y)dAn(Y)dY, n-700

where the convergence is uniform. Using proposition 3.4 we get g is a conti-nuous function.

Since PP

A1 ~ PAl and the random walk is conservative i t follows from theorem 1.2 that g PAl A-a.e. By proposition 1.3 we get that g takes values one or zero A-a.e. Since g is continuous g takes values one or zero only. Since G is connected g is identically one or zero. By theorem 1. 1 we have g = 11.-a.e. on A and therefore g = 1 everywhere on G.

We now have 1 = g <_- P 1_A is Harris.

~ 1. Hence PAl = 1 everywhere on G and therefore P

o

Remark: Some condition like the connectedness of G in theorem 3.1 is neces-sary, which can be illustrated by the following example. The identity on~ is a recurrent random walk, which is spread out but not Harris.

A random walk P on (G,L) is said to be ergodic if for every A E L with

A(A) > 0 the probability that the random walk will visit the set A is posi-tive for A-almost all starting points i.e. PAl> 0 A-a.e. for all A E L with

A(A) > O. Usually ergodicity is defined by the fact, that every invariant set or its complement is empty mod the measure under consideration. It is easily verified that our definition is equivalent with the latter definition.

Theorem 3.2. A random walk P on (G,L) with law p is ergodic if and only if

the random walk will visit every non-empty open set A with positive proba-\'00 *n

bility when started in the point zero i.e. L

n=l p (A) > 0 for all nonempty open sets A.

*n

Prpof. First suppose there exists a nonempty open set A, such that p (A) = 0 for all n ~ 1. Since

00 ₀₀

L

n=l

I

*n 1 A(y)p (dy) = 00

L

n=l p*n(A)

o ,

we have PAlCO) = O.

Choose E > 0 such that _n

I

I

_{A n (A +}T)) has a nonempty T) <E

be a nonempty open subset of n

I

I

A n (A +T)) • T) <e:

For all x E U (0) we have by proposition 2.1

e:

P l(x) = P T l(x) = T P l(x) = P 1(0) ~ PAl (0)

B B -x -x B-x B-x

we conclude, that P is not ergodic, since A(B) > 0 and P

Bl = 0

Conversely, suppose for every nonempty open set A we have I:=l

Let A E E with A(A) > O. In order to show that P is ergodic i t

o

suffices to

show that for all FEE with 0 < A(F) < 00 there exists an integer n such

n *n

tha t P 1A (x) = p (A - x) > 0 on a subset of F wi th posi ti ve Haar measure

Le.

Using Fubini's theorem we get

f

I

F(x)

(J

lA(x+y)p*n(dY»A(dx)

J (J

I

F(x)lA(X+ Y)A(dX))p*n(dY)

Since I

F E £1 (A) and lA E £oo(A) i t follows from proposition 3.4 that

ep(y) =

T

I

F(x)lA(x+ Y)A(dx) is a continuous function. From

J

ep(y)A(dy) = A(F)A(A) > 0 we conclude that {z E G

I

ep(z) > O} is a

non-J

*n

empty open set. Hence I

F(x) p (A - x) A(dx) > 0 for some integer nand

therefore P is ergodic. 0

If a random walk P is conservative and ergodic, then i t follows from

propo-sition 1.3 that for every A E E with A(A) > 0 we have PAl = 1 A-a.e. Hence

Harris recurrency of a random walk is stronger than conservativity and er-godocity. We end this section with an example of a random walk, which is conservative and ergodic but not Harris.

Consider the random walk P on CIR,E,A) with law p, determined by

1

p({n}) = 1 + n and p ({-1 }) = ~l~+~n~n~

Since n is irrational the set A = {nn - m n E:IN, m E:IN} is dense in]R and

i t follows from theorem 3.2 that P is ergodic. Since

J

xp(dx) = 0 i t follows

from theorem 4.5 that P is conservative. On the other hand, A(A) = 0 and *n

p (A) = 1 for all n and therefore the random walk is not spread out and by

4. Recurrence criteria for random walks onmn

In this section we shall consider random walks on ORn,L,A) with n ~ 1, where L is the a-algebra of the Borel sets ofmn and A the Lebesgue measure on

ORn,L). The theorems and propositions of this section are slight modifica-tions of corresponding theorems in [8J. The techniques used in this section for random walks onmn can also be used for random walks on locally compact groups. We start with some notations. For all x and y inmn we write

(x,y) n

I

X_iY_i i=l

I

x

I

= I(x,x) d(x,y) = Ix - yl

Proposition 4.1. Let P be a dissipative random walk on ORn,L,A), then for

I

oo n n

every bounded set A E L we have 0 P 1 is bounded onm •

n= A

Proof. Put U = {x

_I

_Ixl < 1}. Because of the corollary of theorem 2.2 there exists a point p E U such that P

ul (p) s < 1. Choose e: > 0 such that U2e:(p) c U. For every q E U (p) we have by proposition 2.1

e: P_{u ( )}T 1(q) p p-q e: Hence P 1(p) :5 U_2e: s < 1 • s .1_u (p) • e: We shall prove pn 1 Ue:(p) n n-l Suppose P_u (p)l :5 s e: n-l

:5 s for n ~ 1. For n = 1 the assertion is true.

for some n > 1. Then

pn+1 1 U (p) e: n n-l We conclude P u_e:( )_p 1 :5 s for n ~ 1. Hence 00

I

P~

(p)l :5 1

~

s n=l E n onm 1 n :5 1 on m • By propos i-- s for all a E JRn.

From proposition 1.2 we conclude

I

oo=l pn1 ()

00 n n 1 Ue: Pn

tion 2. 1 we have \' P 1 < .. .. on "TO

Ln=l Ue:(p)+a - 1 - s ~

Since A can be covered by a finite number of translates of the hypersphere U (p) the proposition has been proved.

proposition.4.2. every open set A

n

Let P be a conservative random walk on OR ,I, X), then for \'00 n

we have Ln=O P l A = 00 on A. Proof. From theorem 2.2 we conclude PAl have pnl = 1 on A. Then

A

1 on A. Suppose for some n > 1 we

n PAl

=

1 on \'00 n Ln=l P l A We conclude 1.2 we have

A for n

~

1. Hence I:=l

P~l

=

=

00 on A.

00 on A. By proposition

o

We now introduce a class T of functions on JRn. A function f on JRn belongs to T if the following conditions are satisfied.

1) f is a nonnegative continuous even function on En such that f(O) > O. 2) There exists a nonnegative nonincreasing function ~ on the nonnegative

real numbers such that

i) f(x) $;~(lxP forallXEJRn ,

ii)

1

~(lxI>X(dx)

< 00 • JR

n

Theorem 4.1. Let P be a random walk on OR ,E,X). The random walk is conser-vative if and only if there exists a function f E T such that

00

I

n=O

00 •

Proof. Suppose the random walk is conservative. Let A be an open set around zero such that f ~ ~f(O)lA. Then we have

00

~f(O)

I

pnl A(0) n=O

From proposition 4.2 we conclude \'00 0 pnl (0) = 00 and therefore \'00 0 pnf (0) =00.

L_n

=

A Ln

=

\'00 n

Conversely, suppose Ln=O P f(O) = 00 for some f E T. Let $ be a function as-sociated with f as described in the definition of T. Define h(x)

=

$(Ixl) for all x E JRn, then we have I~=o P~(O) = 00. Put

n

K = {x E JR

I

0 $; xi < 1 for i

=

1,2, ••• ,n} • There exists a countable partition of JR~K

1,K2, •••} such that Ki is a trans-late of K for all i. Define the function k on JRn

k (x)

I

_{sup h (Y) 1} K. (x) i=1 YEK i ~ n for all x c E

It is easily verified that k (x)

s

1jJ(

I

x

I -

In)

for

I

x

I

~

In

and therefore

f

k(x)A(dx) < 00. Since h(x) s k(x) we have

L:=O

P~(O) = 00. Suppose the random walk is dissipative, then we conclude from proposition 4.1 and 2.1 that there exists a number w such that

00

L

pn1K s w for all i ElN

.

n=O ~

We now get

00 00 00 00 00

1: p~(x) 1: pn

L

I

hey)

L

n

sup hey) 1_K. (x) sup p l

K. (x)

n=O n=O i=1 YEK. ~ i=1 YEK

i n=O ~ ~ 00 s w 1: sup hey) i=1 YEK i w

f

k(x)A(dx) < 00 •

Hence 1::=0 P~(O) < 00. This contradiction proves, that the random walk is

conservative.

o

Proposition4~3. Let P be a random walk on ORn,E,A) and define the function

n f on E by 2 n sin x. f (x) = 11 ~)n for all x E lRn 2

.

i=1 x. ~

The random walk P is cons erva ti ve if and only if

Proof. It follows from theorem 4.1 that i t suffices to show f E T. Since . 2 sin2x

for all x E lR we have s~n x + 2 s 2 we have

x Hence . 2 s~nx s 2 x 2 2 1+x for all x EE . f(x)

=

n II i=1 n II i=1 (. 1 )n 2 1 +x. ~

The function f satisfies the two conditions in the definition of the class

o

for r ~ 0 . ~(r)

T with the function ~ given by

2

2n

Let p be a probability on ([Rn,L;), then the characteristic fu.nction p of p is defined by

for all t E: IRn •

-For every function f E £1(A) the Fourier tran~form f of f is defined by

f (t)

f

f(x)e-i(x,t)A(dx) for all t E: IRn •

Proposition 4.4. Let P be a random walk onIRn with law p and let f E: T such that

f

If(t) IA(dt) < 00 then

00

. 1. lim

f

f

(t)Re 1 _

r~

(-t) dt •

(2'IT)n rtl

Proof. Suppose f E T and

f

If(t) IA(dt) < 00 We have

/"....

Pf(t)

f

e-i(x,t)Pf(x)A(dx)

J

e-i(X,t)(f f(x+y)p(dy))A(dx)

f

ei(y,t)

(f

f(x+y)e-i(x+y,t)A(dx))p(dy)

=

J

ei(y,t)f(t)p(dy) Hence

~

=

f(p)n for n

~

O.

Choose 0 < r < 1. Then we have

f

e-i(x,t) n=O

!

r~nf(x)A(dx)

00

I

n=O

00

I

nn

(The series r P f(x) converges uniformly and is integrable.)

~

f Hence L O r P f_n= = 1_{- rp}~.

Since f is a bounded continuous function, i t follows from the dominated con-vergence theorem, that pnf is continuous for all n. Hence

I:=o

r~nf

is a continuous function such that its Fourier transform is absolutely integra-ble. From a well known inversion theorem (see e.q. [4J, p. 2) we conclude

I

r~nf(x)

1

I

ei(x,t)f(t) .

l~

dt

n=O = (21T)n 1 - rp (t) for all x

E lRn •

Since f is an even function the function f is real valued and we get

00

I

n=O pnf(0) = :.-...;;.;1... lim (21T) n r t l

f

-

1 f (t) Re 1 _ rp (t) dt •

o

Proposition 4.5. Let p be a probability on ORn,~) with characteristic func-tion p. If for some t E:IRn we have p(t) = 1, then

p

is periodic with period t. Proof. Suppose p(t) = 1. Then we have

p(t)

J

cos(x,t)p(dx) + i

f

sin(x,t)p(dx) = 1

Hence cos(x,t) = 1 and sin(x,t) = 0 for p-almost all x ElRn and therefore

i(x,~ n

e = 1 for p-almost all x E lR • Hence

p

(x) •

o

Proposi tion 4.6. Let p be a probability on ORn I~) with characteristic func-tion p. If there exists a number w > 0 such that

lim

I

Re 1 _ :, ( ) dt 1= 00

rt1 rp t

I

tl~w

then for all a. > 0 we have

lim

f

Re 1 _

1~

( ) dt 1= 00 •

rt1 rp t

Itl~a.

Proof. Suppose there exists a number w > 0 such that

lim

I

rt1

Itl~w

and let a. > O.

1

There exists a sequence r t 1 (n + 00) such that n lim

I

Re ..

1~

.... dt

=

t < 00 • 1 - r p (t) n~ Itl~w n 1 lim

I

rtl B

~

a. and x

~ u

k

j

=

1 U (a.)}. Sincew J B is compact and

< 1 on B we conclude from the dominated convergence theorem that

Ipi

compact and there exists k

that A cu. 1 U (a.).

J= w J

{x E: En

I I

x

I

Put A

=

{x E: En

I I

x

I

~ a. and

p

(x)

=

1}. Since p is continuous the set A is a finite number of points a

1,a2, ••. ,ak in A such Put B

Using proposition 4.5 we conclude lim sup rtl

f

Re

l~;P(t)

dt

~

kt + s < 00

I

t

I

~a. and therefore lim rtl

f

Re l-;P(t) dt

-F

00 • Itl~a. IJ

Theorem 4.2. Let P be a random walk on (IRn,E,A) with law p. The random walk is conservative if and only if there exists a number a. > Osuch that

lim

f

Re rtl \t

I

~a.

1 - rp(t) dt 00 • Proof. Let f be Put Q {x E: En ~ n n l Q(t) 2 Hi =l rier transforms

the function as defined in proposition 4.3.

I

Ixil ~ 1 for i

=

1,2, ..• ,n} then the Fourier transform sin x./x .. It follows from the convolution theorem for

Fou-~ ~

that

~ 2

1*2n

=

4- n f

Q

and from a well known inversion theorem ([4J, p. 2) that

1*2n(x)

=

4-n2

.

f

f(t)ei(x,t)dt •

Q (21T)n

2

Hence f(t)

=

4n (21T)n 1*2n(_t) and therefore f

~

0 and has compact support.

Q

Hence there exists a number w > 0 such that f(t)

=

0 for all It

I

> wand we conclude from proposition 4.4

00

f

I

pnf(0)

₌

1 _{lim f (t) Re} 1 _{dt •} rp(t) n=O (21T) _{n rtl} 1 -Itlsw

First suppose there exists a number a. > 0 such that

lim

_f

Re . 1 dt

rp(t) 00 ,

rtl 1

-Itl~a

then i t follows from proposition 4.6 that for all S > 0 we have

lim r Re 1

-~P(t)

rtl

Iti~S

dt 00

Since f(O)

#

0 there exists a number v with 0 < v ~ wand f(t) ~ ~f(O) for all It I ~ v. We then have

00

- 1

f

1

~ ~f(O) . .. Re ...,;., . . dt=oo.

(2 )n 1 - rp (t)

1T

I

t

I

sv

From proposition 4.3 we conclude the random walk is conservative. Converse-ly, suppose the random walk is conservative. Then by proposition 4.3 we have

I:=o

pnf(O) = 00. Put M = max f(t), then we get

Itl~w Hence 00 00 =

I

n=O pnf (0) ~ ~M_ lim inf (21T)n rtl

f

1 Re ~l--~rp~(t-)- dt • Itl~w lim

J

_Re 1 dt 0 rp(t) 00 rtl 1 -Itl~w Corollary. If

f

Re 1 _ 1 dt

P

(t) = 00 Itl~a.

for some a. > 0, then the random walk with law p is conservative.

Proof. From Fatou's lemma we conclude

Hence

J

Re 1

_l~(t)

dt

~

I

t

I

~a. lim inf rtl

I

Re "7"l....

_-'-'-~..,.;~r.(-t":'")

dt • Itl~a.

lim rtl

I

R 1 e 1 - r~(t) Itl::;a dt (J()

and therefore the random walk is conservative.

o

A probability p on ORn,~) is said to be ~educible if there exists a linear

n

subspace L ofm such that dim L < nand p(L) = 1. A probability which is

not reducible is called irreducible. A random walk with law p is called ~­

reducible if its law p is irreducible.

Proposi tion4.7. Let p be a probability on ORn,~) and define for all a ;::: 0

n the function </> on m by a </> (x) a

I

(x,y) 2p (dy) IYI::;a for all x E mn •

The probability p is irreducible if and only if there exists a number a > 0

such that </>a > 0 on {x I Ixl = 1}.

Proof. First suppose p is reducible, then there exists a linear subspace L

of mn with dim L < nand p(L) = 1. Let y E mn with Iy

I

1 and y orthogonal

to L. Then </>a(y) = 0 for all a > O. Conversely, suppose p is irreducible.

For all n EIN put W_n = {x E mn

I

Ixl = 1 and </>n(x) = A}. If n:=l W_n

f:.

~,

then there exists a point y E mn with Iyl = 1 and

f

(z,y)2p (dy) = O. It

fol-lows p({x

I

(x,y) = O}) = 1 and hence p is reducible. This contradiction

(W ) n

o

</> > 0 on {x

I I

x

I

= 1}. m W n

continuous the set W is compact and hence

n

~. Since the sequence

(J()

proves n_n=1 W_n =~. Since ~_~n is

m

there exists a number m EIN such that nn=l

is nonincreasing we conclude W =~. Hence

m

Proposi tion 4.8. If P is an irreducible probability on ORn,E) with

characte-ristic function

p,

then there exist positive numbers Aand ~ with

~

I

2

Re(l - p(t» ;::: A tl for It

I ::;

~

•

Proof. Since p is irreducible i t follows from proposition 4.7 that there

exist positive numbers a and cr such that

for Iy

I

= 1 •

2

(x,y) p(dx) ;::: cr

J

Ixl::;a

Isin

~(x,t)

I

~

J

(x,t)

I

1T i f Ixl

~ l~1

.

Choose t E lRn such that 0 < It I ~ 1T We now have

ct Re (1- p(t»

=

2

f

sin2~(x,t)P(dx) ~

2

r

sin2~(x/t)p(dx)

Ixl$l~r

Hence

~

1

f

(x,t>2p (dX);

l

f

(x,t)2p (dx) 1T 1 T .

Ixl~~

Ixl~ct

21t l2

f

(

t)2 (d )

~

_{20 ltl 2 •} =

7

X'N

p x 1T Ixl$ct ~ 20I 12 Re(1 - P (t» ~ -"- t 1T for It I

o

n

Theorem 4.3. If n ~ 3 , then every irreducible random walk on

OR

,E,A) is

dissipative.

Proof. Let P be a random walk with law p. We have for 0 < r < 1 1 ~e(1-

rp

(t) ) Re(1 ~ rp (t) ) Re = $ 1

-

rp (t) 2 ~ 2 11 - rp(t) I (Re (1 - rp (t) ) .1 1 Re(1

-

rp (t) ) ~ Re (1 - p(t) )

.

f

dt

~ltl2

• Itl~J.l

~

Altl2 for all It

I

$ ].1. Hence

J

1 1 dt $ Re(1 - p(t» dt~

'r

Itl$J.l 0<

J

Re . 1. .. - 1 - rp(t) \tl$J.l

From proposition 4.8 we conclude, that there exist two positive numbers A and ].1 such that Re(1 - p(t»

Since we conclude lim rtl 1 dt

t:

00 - rp(t)

and therefore by theorem 4.2 the random walk is dissipative.

o

Because of theorem 4.3 the situation with respect to conservativity of ran-dom walks onlRn if n ~ 3 is quite simple. Things are less simple for

ran-2

Proposition 4.9. Let p be a probability on ORn,E) with

J

jxlp(dx) < 00, then

Proof.

~~

Re (

lit'

(t» = 0 •

Re(l-p(t»

=

2

f

sin2~(x,t)p(dx)

S;21tl

f

Ixllsin2~(x,t)lp(dX)

• (x,t)

.

ISin2~(x

t)

I .

S~nce '(x,t)' , ~s bounded and tends to zero if t ~ 0 we conclude from the dominated convergence theorem that

o

Proposition 4.10. Let p be a probability on ORn,E) such that

J

Ixlp(dx) < 00 and cr2

J

Ixj2p(dx) < 00, then

i) Re(l - p(t» S;

~cr21t12

,

ii) if

f

xp(dx)

=

0, then lIm p(t)

I

S;

~cr21t12

•

Proof. Re(l-p(t» =2

J

sin2~(x,t)p(dX) s;~ltI2

J

jxI2p(dx) =

~cr21t12

which proves i). sin(x,t)

=

(x,t)-~(x,t)2sin(e(x,t»

where e depends on

(x,t) and O,<e < 1. Hence

Im p(t) =

J

sin(x,t)p(dx)

=

f

(x,t)p(dx) -

~

J

(x,t)2sin (e(x,t»P(dx). From

J

xp(dx) = 0 we conclude

J

(x,t)p(dx)

=

(t,J xp(dx»

=

0 for all tEIRn•

Hence IIm

p

(t)

I

S;

~i

It 12• 0

Proposi tion 4.,,1 L Let p be a probab ili ty on OR, E) such that

J

Ix

I

p (dx) < 00. If we put ~

=

J

xp(dx) then

lim

~mp(t)

=

~

•

t~ t

Proof. rm p(t)

=

I

sin(x,t)p(dx)

=

t

f

x

Si~txt

p(dx) From the dominated convergence theorem we conclude

lim

I~P(t)

=

~

.

t~ t

Proposition 4.12. Let p be a probability on CIR,E) such that

J

\xlp(dx) < 00 then

1

I

Re(1 -1

-

pet)~ t2 dt < 00 • Proof. Define 1

ep(x)

I

₁ - cos .xt dt for all x E lR •

2 0 t Then 1 x2n+2t2n

f

00 <I>(x)

L

(-1) n dt • (2n + 2) ~ 0 n=O

The uniform convergence of the series permits interchanging of summation and integration and we get

00 <I> (x) =

I

n=O 2n+2 (-1)n x (2n+2) ~(2n+1)

and therefore </>" (x) = sin x

x

I

<1>' (x)

I

~

1

and <I>(x) ~

I

x 1

f

Re (1 -1

J

OO sin t dt 1T

• Since <1>' (0) = 0 and 0 t =

2

we have for all x E lR. Hence

1 00

J

:i

f

(1 - cos xt) p (dx)

-1 -00

00

f

<I> (x)p(dx)

~If

Ixlp(dx) < 0 0 .

D

2 Theorem 4.4. Let P be an irreducible random walk with law p onlR orlR such that

J

Ixlp(dx) < 00 and 02

=

J

IxI 2p(dx) < 00. If

~

=

J

xp(dx) = 0, then the random walk is conservative.

Proof. From proposition 4.8 we conclude that there exist positive numbers

Aand p such that

if

I

tl ~ p •

Since ~

o

we conclude from proposition 4.10

~41~14

(Re(1 - p(t») 2

~ ~

i:lt.L.

4 4

~ 2 0 t

Hence if It

I

~ p we have Re(1 ~

p

(t) ) ~ -,-2_A--:-~ 2 ~ 2 ~4ItI2' (Re(l-p(t») + (1m p(t» v 1 Re 1 - p(t) = Therefore

f

Re

~~1,...,.-

dt 1 -

P

(t) Itl~p

>.?2.

f

~='"

- 4 2 ' a

Itl~p

It

I

From the corollary of theorem 4.2 we conclude, that P is conservative. 0

Theorem 4.5. Let P be a random walk onJR with law p such that

J

Ixlp(dx) <"'. The random walk P is conservative if and only if ~

=

f

xp(dx)

=

o.

Proof. First suppose ~

=

O. Given any E > 0 i t follows from the propositions 4.9 and 4.11 that there exists a number a > 0 such that 11m p(t)

I

~ Eltl and IRe(l - p(t»

I

~ Eltl if It I ~ a. For 0 < r < 1 we have

1 1 - r

Re ~

~--""'""""'-"""""_::""""--':=--~-_"""':"'--'-'-'--:-1 - rp(t) 2 2 ~ 2"

(Re(1 - rp (t) ) ) + r (1m p (t) )

It is easily verified, that for 0 < r < 1 and 0 < a < 1 we have

2 2 2 2

(1 - raj ~ 2(1 - r) + 2r (1 - a)

Therefore we can choose a small enough such that

(Re(1 - rp (t) ) )~ 2 ~ 2 (1 - r) 2 + 2r (Re (1 - P t»)2 ~(2 for It

I

~ a • We now have a a

f

Re 1 dt

J

(l ,.. r)dt 1 - rp(t) ~ ~ 2 ~ 2 (Re(1

-

rp (t) ) ) + (1m p(t» -a -a a

_._._.

a 1-r (l - r)

J

dt 1

J

dt ~ > -2 _{3r2E2t2 - 3 1+~2t2}

.

2(1-r) + -a

_-

_._.-

a Hence 1-r a lim

J

Re 1 dt ~ L rtl 1 - rp(t) 3E -a and therefore a lim

J

Re 1 dt rp(t) '" rtl 1

--a

and by theorem 4.2 P is conservative. conversely, suppose ~ ~

a

1 Re 1 - r~(t) = Re(l :-

P

(t) + (1 :- r) Re

p

(t) .. ~ 2 2 ~ 2 (Re(l - rp (t) ) ) + r (1m p (t) ) :;; Re(1 .:-

P(

t)) + --,-,-_l,--,--~r_~ r2(lm p(t»2 11-rp(t) 12 By proposition 4.11 we can choose a >

a

such that

I

1m

p

(t) I

~

I

~2t

1 for It1 :;; a . Hence -a a

f

Re 1 dt :;; 4 2. 1 - rp(t) ~ dt •

By proposition 4.12 the first integral on the right hand side is finite. We are now going to estimate the second integral.

1 - r dt \1 - rp(t) 1 2 t :-r dt 2 ~ 2 r (1m p (t) )

J

1-r:;;lt!:;;a

J

dt2= 2+8(l :-r)2 2

(..J..._1)

1-r a 1-r:;;lt l:;;a t r ~ dt ~ + 1 - r

f

~dt+1 - r

J

It l:;;l-r \l-rp(t) 1 2 1-r:;;!tl:;;a

f

It!:;;l-r :;; 2 + 4(1 - r ) 2 2 r ~ a

J

1 -r -~~--::- dt \1 - rp(t) 1 2 -a Hence a lim

J

Re 1 _

~P(t)

dt

~

00 rtl -a

and therefore by theorem 4.2 the random walk is dissipative.

o

References

[lJ Breiman, L.: Probability. Reading, Massachusets: Addison-Wesley 1968. [2J Chung, K.L. and W.H.J. Fuchs: On the distribution of sums of random

va-riables. Mem. A.M.S. nr. 6, 1-12 (1951).

[3J Foguel, S.R.: The ergodic theory of Markov processes. New York: Van Nos-trand 1969.

[4J Reiter, H.: Classical Harmonic Analysis and locally compact groups, Oxford University Press, 1968.

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

[6J Simons, F.H~ and D.A. averdijk: A recurrent dissipative Markov process. Internal publication

casaR

77-02. Eindhoven University of Techno-logy, 1977.

[7J Simons, F.H. and D.A. averdijk: Recurrent and sweep-out sets for Markov processes. Mh. Math. 86, 305-326 (1979).