On compact stochastic perturbations of mappings of the unit internal

Problem y M atem atyczne 13 (1992), 13-24

On compact stochastic perturbations

of mappings of the unit interval

W ojciech Bartoszek

Let <pn b e continuous m appins o f a com p act m etric space X and 3? b e som e fixed co m p a ct M arkov operator on C ( X ) . W e study the a sy m p totic ( n — * o o ) beh avior o f invariant probability measure o f the com p osition s ( TVnR )*, where <pn — * ip uniform ly. W e apply our generał results to the investigation o f the difference eąuation X n+\ — <p(Xn) + W , where W is a fixed random variable independent o f n and X n and <p are continuous m aps from [0 ,1] into [0 ,1].

It is shown that for a w ide class o f m appings <p this M arkov process adm its the uniąue (station ary) invai'iant measure p(<p) and the m apping (p — > p(<p) is continuous.

Let ( X , p ) b e a com p a ct m etric space. W e denote by C ( X ) the B anach lattice o f all continuous functions on X , and by P ( X ) the set o f all B orel p rob a bility measures on X . T h e sm allest, closed set o f all fuli m easure p € P ( X ) is denoted by supp p (th e support o f p ). A linear op erator T : C ( X ) — * C ( X ) is said to b e M arkov if T l = 1, and / > 0 =>• T f > 0. It is well known that for every M arkov operator T on C ( X ) there exits a uniąue fam ily o f probability measures P ( x , •), on X such that

(a ) for every B orel set A the m apping x — > P ( x , A ) is B orel m easur- able

In fact, we have P ( x , - ) = T*8X(-). B y B ( X ) we denote the set o f b ou n ded Borel functions on X . Using prop erty (b ) the M arkov operator T can b e canonically extended to an operator T : B ( X ) — * B { X ) . W e say that a closed set A C X is T -in va rian t (or sim ply, invariant if T is fixed) if for every x £ A we have P ( x , A ) = 1 (equivalentely T \ A >

1^)-N ow, let fi be the p rod u ct space II^Lo %n where X n = X . W e eąuip fi with its natural to p o lo g y and p rod u ct cr—field. Let i]n b e the natural p rojection r]n : fi — > X n. It is well known (e.g. [3] P rop osition 2.10 p.18) that given any initial distribution probability /z on X there is a probability V M defined on fi with

Vli{r)keAk:k = 0,l,...,n} = J

1

AoT(l

Al... (:

T\An).. .)dfi.

M oreover, the seąuence ’s a hom ogenous M arkov chain with transition p robability P ( •,•) and starting m easure /j,. It is called the canonical M arkov chain with transition probability P .

Let P j ( X ) denote the set o f all T *-invarian t measures. Clearly, it is nonem pty, con vex and iu *-com p act subset o f P { X ) , and for every /z belongin g to Pt{ X ) the canonical M arkov chain {//;-} is stationary with respect to 7?M.

In the sequel we will need som e inform ations on com p a ct M arkov operators. R ecall that a linear operator T : C ( X ) — >• C ( X ) is com p a ct if and on ly if the m apping x — > T*8X is norm continuous, so every c o m ­ pact M arkov operator is strong Feller ( T B ( X ) C C ( X ) see [3] P rop osi­ tion 5.8 p.37 ). T h e Cesaro means A nf = n -1( / + T / - ) - . . . + 71'1-1/ ) o f a com p a ct M arkov operator T converge uniform ly to a finite-dim ensional p ro je ctio n , ex Pt( X ) (it means the set o f extrem al m easures) is fi- nite and supp fls u p p /Z2 = 0 for each distinct extrem al T *-invarian t measures /Xi, /tj (see [2] and [4] for detailes). O bserve that every con tin ­ uous m apping 9? : X — > X defines a M arkov (d eterm in istic) operator Tv f = f o (p.

Now let M denote the con vex, sem itop ological sem igroup o f all M arkov operators on C ( X ) . For an op erator R € N the m apping M 3 T — > T o R is denoted by R . O bserve that if T'v is a de­ term inistic M arkov operator induced by som e transform ation p then K { T „ ) f { x ) = Tv o R f ( x ) = R f ( i p ( x ) ) = f f ( y ) R ( < p ( x ) , d y ) . If for ev- ery x 6 X the transition m easure R ( x , - ) is con cen trated on the bali

On Co m p a c t St o c h a s t i c Pe r t u r b a t i o n. 15

B ( x , r ) = { y G X : p ( x , y ) < r } , then our operator 7^(7^) can be recognized as an r-p e rtu rb a tio n o f the dynam ical system ( X , ip). In the seąuel we will consider only com p a ct perturbations (thus 7Z (T ) is a com p a ct for every T G M ) . Clearly, 7Z(Tn) 7Z(T) whenever Tn T (here s.o.t. means strong operator to p o lo g y ). Using the com pactn ess o f the operator R we get the follow ing.

L e m m a 1 Let the seguence (Tn) o f Markov operators on C ( X ) con- verge in the strong operator topology to T and R be a fixed compact Markou operator on C ( X ) . Then 7Z(Tn) — * 7Z (T ) in the operator norm and every limit measure p = w* — lirn, >oc p n is 7Z (T )* inuari-ant (here p n G Pn(Tn) ( X ) ) . Moreouer, in this case || p — p Uj ||— > 0, where || • || is the uariation.

P r o o f . B y the com pactn ess o f R we can chose a finite, £ -d en se subset f i i - , fm o f R ( K i ) , where 7\i denotes the unit bali o f C ( X ) . Since Tn T thus there exists no such that for n > no we have || Tnf j — T f j ||< £ for every j = 1 , . . . ,m . So, for every / G C ( X ) , || / ||< 1 we get

|| R ( T „ ) / - K(T)f||<|| - || +

+ II K ( r „ ) / , - 7 Z(T)fiII + II ~ ||< 3£ if the index j is suitable. Thus,

|| 7 l(T n) - 7Ł(T) ||= sup \\TL{Tn) f - T l [ T ) f \ \

ll/ll< i

tends to 0.

Now assume that p — w* — lim^ ,00 p U] for som e p Uj € Pn(Tni) { X ) -For every / G C ( X ) we have

\ J f d p - j n ( T ) f d p \ < \

J

j

|

J

J

J

J

II P - Pn, ||= sup | I f dp - I f dpnj =

sup | [ K ( T ) f dp - [ H ( T nj) f dpnj |< |/||<i

J

J

J

J

J

(here { / i , . . . , f m} is an e -d e n se subset o f R ( K i ) ) . Since e can be taken arbitrarily sm ali, then || p — p n. ||— > 0.

T h e follow ing corollary strengthens som e results from [lj.

C o r o l l a r y 1 I f the M arkov operator 7Z( T) has exactly one invariant probability measure p (i.e. it is uniquely ergodic ) and Tn T then || p n — p ||—> 0 (here p n G Pv.(Tn) ( X ) ) . In particular there exists n 0 such that f o r n > no the operators 7Z(Tn) are uniquely ergodic.

N ow, let ip be a continuous m apping from the unit interval [0,1] into itself. C onsider the stochastic difference eąuation X n+i = <p{Xn) + W , w here W is a sm ali random variable possessing the p rob a bility density fu n ction g : [—a, a] — > [0, o o ), where a is sm ali and g > 0, / “a g d\ = 1 for the Lebesgue m easure A (i.e. P r o b . ( W E A ) = f A g ( x ) d x for every Borel set A C [—a, a] ). W e assume that the p ertu rbation term W is independent o f n and X n.

W e say that a closed subset ([1] D efinition 1) S C [0,1] is invariant with respect to our stochastic difference eąuation if <p(S) ® [—a, a] C 5 , w here © is defined A ® B = { x -\- y : x G A , ? / G B } . W e will say S is (<p, a) invariant then. Let us observe that every (<p, a) invariant subset S is invariant in an ordinary sense (i.e. <^(5') C S ). If there exists a ((p, a) invariant set S then clearly X n is a M arkov process w ith state

On Co m p a c t St o c h a s t i c Pe r t u r b a t i o n. 17

space S. M ore precisely, for every initial probability g concentrated on S there exists the M arkov process X n such that

V , ( X n+1 - <p(Xn) e A ) = [ g d \ J A

for every B orel A C [—a ,a ]. W e show that the process X n defined on som e (ip, a) invariant subset can be regarded as a part o f the M arkov p erturbation o f the dynam ical system ([0 ,1 ], <*£>). W e set

g (t — x ) , for x € [a, 1 — a]

d R*óx a _2(a — x ) • lf0,a)(^) + a ~1 • x • g ( t — a ),fo r x £ [0, a) [C) d X

« - 2( i - ( l - a ) ) l M

( i) +

a _1( l — x ) ■ g (t — (1 — a )), for x € (1 — a, 1] where A denotes the Lebesgue measure. Clearly, the m apping

x — ■> R* 8X

is n orm continuous, so the M arkov operator R given by the above tran­ sition fu n ction is com p a ct.

M oreover, sińce R*6X -<-< A then R * ( P ( S ) ) C ^ ( S , A) and it is easy to see that

= / ~ ^ f o r e v e r y ^ G p

W e n otice that if g is a function o f bou n d ed variation then sim ilarly d j has bou n d ed variation too, and the follow ing rough estim ation

V a r ( - ■■ x ) < V a r ( g ) + 2a~l d A

holds.

C onsider the perturbation o f Tv by the M arkov operator R and let gn b e the canonical M arkov process defined by 7Z(TV). For every B orel set A C S and x £ S (n otice that (p(x) € [a, 1 — a] then ) we have

P(Vn+1 G A | gn = x ) = 7l ( T v )\A( x ) = ( R o TV)*8X{ A ) =

= R*8v(x)( A ) = f g ( y - <p(x)) d \ (y). JA

On the other hand if x £ S then

P(xn+

e A\xn =

P(<p(xn) + w e A\xn =

= / , 9{ y ) d \ { y ) = / g ( y - p { x ) ) d \ ( y ) .

J A - t p ( x ) J A

Since X n and gn are M arkov processes we con clu d e that all finite di- m ensional distributions coincide. So on the phase subspace S they give equivalent descriptions o f the perturbation o f the dynam ical system ([0,1], v )

-A ssum e that ip is nonsingular (i.e. if -A(y4) = 0 for a B orel set -A C [0,1] then A(<^-1(/1 )) = 0 ). T h e Frobenius-Perron op erator con n ected w ith p is an operator defined on T x([0,1]) as follows

In [1] Boyarski has studied the stochastic operators Q v defined on T1(S') b y the eąuation Q u>f { x ) — ( P v f * g ) ( x ) , where * denotes the convolu- tion and g is the probability density fu n ction o f W . T h e properties o f invariant densities have been investigated there. In particular it was shown that the ” regularity” o f g im plies som e nice properties o f invari- ant density. Since for a ( p, a) invariant set S the operators Q v and TZ(TV)* coin cid e on T1(5') (n otice that S is an 7Z(TV) invariant subset th en ), thus { / 6 L 1( S ) : Q v f = / } = Pn(Tv)(S)- T h e existen ce o f an absolutely continuous invariant m easure o f is a sim ple conseąuence o f the last eąuality. M oreover it does exist for arbitrary continuous m appin g (n ot necessary nonsingular or C1 ) from [0,1] into itself. C o r o l l a r y 2 I f f o r a continuous mapping ip : [0,1] — » [0,1] there exists a closed ( p , a) inuariant set S then f o r euery random uariable W with absolutely continuous density function g : [—a, a] — > [0, oo) the stochastic process defined by Xn+1 = p ( X n) + W has a stationary probability distribution o f the fo r m V j f o r som e positiue, normalized f £ T x(5 , A). In particular f o r euery B orel set A C S V f ( X n £ A ) = I A f d X

N ow , let us fix the p robability density fu n ction g o f W and assume that supp g C [—a, a). O ur next result corresponds to T h eorem 1 from [1].

On Co m p a c t St o c h a s t i c Pe r t u r b a t i o n. .. 19

P r o p o s i t i o n 1 Let ip be a continuous mapping fr o m [0,1] into itself and IZ(Tv )be the compact perturbation*pf Tv (here R is defined by (c )) . I f g is o f bounded uariation then the density o f each IZ(T^)* inuariant probability p is o f bounded uariation too.

P r o o f . Let p £ ^72(7V)([0,1]) b e arbitrary. Since

d p [ d 7Zl TuYSi , x s

A w- / —

/■(*).

then for every 0 = t0 < t\ < . . . < tn = 1 we get

£ ! , M x ) < V a r { g ) + 2 o - , < ^ T h e follow ing theorem connects the sm othness o f invariant densities w ith properties o f g.

T h e o r e m 1 Let p be a continuous mapping fr o m [0,1] into itself and S be ( p , a ) inuariant subset o f [0,1].

I f ip is .nonsingular and g £ L °°([—« , « ] ) then density o f euery 'R(TV)* -in u a ria n t probability p supported on S is a continuous f u n c ­ tion.

I f g £ C q ([—n ,a ]) (i.e. g ( —a) = g ( a) — 0 and g has continuous k deriuatiues ) then ^ £ (^ ([O , 1]) f o r each inuariant probability p £ Pn(Tv) ( S ) .

P r o o f . Let p b e nonsingular and p £ P k ( t )(*S)- Since for each t £

[0,1]

- <p( z) ) dp( x) =

j sg(t-

then £ C ( [ 0 , 1]) with supp C S. Now, let g £ C o ( [ 0 ,1]). Then

d

= J

h—*00 h

so dX ^ Crl([0 ł !])• Similarly we get higher derivatives.

R e m a r k In the previous theorem the nonsingularity assum ption on <p is essential. In fact let <f(x) = 2_1 for all x £ [0,1] and

g ( x )

(4 a ) 1 for x £ [—a ,0 ), 3 (4 a) - 1 for x £ [0, a].

where a < 4_1 is fixed. Clearly, the unit interval is (y?,a) invariant and the (uniąue) IZ(TV) invariant p robability g has the density o f b ou n ded variation. But it can b e com p u ted that

d g

= J

_ 9 - i a _ f (4 « )

1

[2

1

, 2

P r o p o s i t i o n 2 Let p be a nonsingular continuous mapping fr o m [0,1] into itself and g £ L 1( [ —a, a] ) be a density o f som e perturbation o f p such that 0 £ suppg.

I f there are no two disjoint non-m eager p -in v a r ia n t subsets o f [ 0,1] then 7Ł(TV) is uniąuely ergodic.

P r o o f . As in the first part o f our theorem 1, if g is 7^.(r¥>)*-invariant probability, then

j ji ł ) = J g ( t

x ) d l i °d\~

and thus

{ ( € [0 , 1 1 : ^ W > 0}

has n on em p ty interior. But the top ologica l support o f every invariant p rob a bility is an invariant subset o f [0,1] (see [4] for detailes). So, because 0 £ supp g, we get that the support o f every 7Z(T,'v)* in- yariant probability is som e (£>-invariant subset w ith n on em p ty interior.

On Co m p a c t St o c h a s t i c Pe r t u r b a t i o n. 21

Since supports o f distinct extrem al invariant probabilities o f a com p a ct M arków op erator are disjoint (see [4] ) thus because o f the assum ptions o f our theorem the operator 7Z( TV) must b e uniąuely ergodic.

Recall that a continuous m apping p from [0,1] into itself is said to b e transitive if there exists a point xo G [0,1] such that the orbit {<£>n(x0)}n>o is dense in [0,1]. T h e follow ing result is a sim ple conse- ąuence o f the previous resułts.

C o r o l l a r y 3 Let p be a continuous, nonsingular and transitiue map­ ping fr o m [0,1] into itself and g G T1([ —a, a]) be the probability density fu n ction such that 0 G supp g. Then the Markou operator 7Z(TV) is

uniąuely ergodic.

T h e follow ing theorem expresses the continuous dependence o f the invariant measure o f the uniąuely ergodic com p act perturbation 7 l(T v ), on the m apping p .

T h e o r e m 2 Let p n — * p uniformly on [0,1] uihere p n, p are con­ tinuous mappings fr o m [0,1] into itself. I f f o r som e positiue a there exists a closed (n on em p ty) (p , a )-in v a ria n t set S and there are no two non -m eager disjoint p -i n v a r ia n t sets then f o r euery probability density fu n ction g G L \ ( ( —co , + o o ) ) satisfying 0 G supp g C [—a, a], we have

where p n, p are TZ(TVn)*, /K ( T V)* inuariant probabilities, respectiuely.

P r o o f . B y our prop osition 2 the M arkov operator 7Z(TV) is uniąuely ergodic. Since p n — » <p im plies TVn Tv , thus by our corollary 1 we get the thesis.

T h e follow in g exam ple shows that in generał the uniform conver- gence <pn — > ip o f continuous m appings from [0,1] into itself does not guarantee the convergence o f a suitable seąuence o f invariant p rob a bil­ ities.

E x a m p l e Let <p b e the continuous m apping given by the follow ing diagram :

C onsider the density g — 8 on [—(1 6 )- 1 , (1 6 )-1 ] and let R be the appropriate com p a ct M arkov operator defined as in (c ). Cłearly A = [(16)~1, 7 (1 6 )- 1 ], B — [9 (16)_1 ,1 5 (1 6 )- 1 ] are the only (ip, (1 6 )_1 ) in- variant subsets o f [0,1]. Thus ^7?(tv)([0, 1]) has exa ctly tw o extrem al m easures concentrated on A and B respectively. N ow, we define the seąuence o f continuous functions f n on the unit interval:

On Co m p a c t St o c h a s t i c Pe r t u r b a t i o n... 23

and consider the seąuence tpn = f n<p (clearly <pn — > <p). Observe that for od d n the m appings ipn have exactly one ((pn, (1 6 )_1) invariant subset and it is contained in [(16)—1, 7 (1 6 )- 1 ]. For even n the m ap­ pings ipn have also exactly one (<^n,( 1 6 ) - 1 ) invariant subset, but it is contained in [9 (16)_ 1 , 1 5(16)- 1 ]. Thus, for every natural n, we have || Hn — fJ-n+i ||= where fij £ Pk(tv )([0,1]) and the seąuence o f mea- sures fin does not converge.

R e m a r k It is easy to observe that by a smali m odifications in the pre- vious exam ple the m appings ip,<pn can be taken to b e sm ooth.

R e f e r e n c e s

[1] B oyarsky A ., Continuity o f invariant measures f o r families o f maps, A dvances in A p p lied M athem atics 6 (1985), 113-128. [2] Lin 0 . , Quasi-compactness and uniform ergodicity o f positire op-

erators, Israel Journal o f M athem atics 29 (1978), 309-311.

[3] R evu z D ., Markou chains, N orth-H olland M athem atical Library 1975.

[4] Sine F ., G eom etrie theory o f single Markov operator, Pacific Jour­ nal o f M athem atics 27 (1968), 155-166.

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