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Continuous Markov Semigroups and Stability of Transport Equations
Journal of Mathematical Analysis and Applications 249, 668᎐685 Ž2000. doi:10.1006rjmaa.2000.6968, available online at http:rrwww.idealibrary.com on
Continuous Markov Semigroups and Stability of Transport Equations 1 Katarzyna Pichor ´ Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland
and Ryszard Rudnicki Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland; and Institute of Mathematics, Silesian Uni¨ ersity, Bankowa 14, 40-007 Katowice, Poland Submitted by Paul S. Muhly Received November 22, 1999
A new theorem for asymptotic stability of Markov semigroups is proved. This result is applied to transport equations connected with diffusion and jumping processes and randomly perturbed dynamical systems. 䊚 2000 Academic Press Key Words: Markov semigroup; asymptotic stability; transport equation; Markov process.
0. INTRODUCTION Markov operators and Markov semigroups are intensively studied because they play a special role in applications. The book of Lasota and Mackey w7x is an excellent survey of many results on this subject. The main problem of the theory of Markov operators is their asymptotic stability, i.e., when there exists an invariant density f# such that P n f ª f# in L1 for 1 This research was supported by the State Committee for Scientific Research ŽPoland. Grant 2 P03A 010 16 and by the Foundation for Polish Science. Part of this work was done during a visit at the University of Pau. The authors thank French᎐Polish Integrated Action Programme Polonium for financial support of this visit.
668 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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every density f. This problem has been recently investigated for Markov operators with a nontrivial integral part w1, 2, 10, 16x. In particular in w16x was shown that if such an operator has a positive invariant density f# and has no other periodic points in the set of densities, then it is asymptotically stable. The purpose of this paper is to provide new sufficient conditions for asymptotic stability of Markov semigroups and apply them to partial differential equations. In Section 1 we strengthen the results of w16x in the case of continuous time Markov semigroups. Namely, we show that if a partially integral Markov semigroup has only one invariant density f# and f# ) 0 a.e., then it is asymptotically stable. The proof of this theorem is based on the results concerning properties of Harris operators w6x. The question of asymptotic stability of positive semigroups can be investigated also by means of methods of spectral analysis. Such results are based on the assumption that there exists a strictly dominant real eigenvalue of the generator of the semigroup. Verification of this assumption requires the analysis of the whole spectrum of the generator, which is technically difficult. Using our result it is sufficient to check that the semigroup has only one invariant density. Moreover, we do not need to prove that the semigroup is eventually compact. In Sections 2 and 3 we apply results concerning asymptotic stability of Markov semigroups to transport equations. We investigate Markov semigroups generated by the equations
⭸u ⭸t
q u s Au q Ku,
Ž 0.1.
where K is a Markov operator, ) 0, and A is a differential operator of the first or second order. Equation Ž0.1. appears in such diverse areas as astrophysics Žfluctuations in the brightness of the Milky-Way w3x., population dynamics w4, 9x, the theory of jump processes w15, 17x, and multistate diffusion w8, 12x. In order to apply our results we have to check that the semigroup P Ž t .4t G 0 generated by Eq. Ž0.1. is partially integral. In Section 2 we consider two such cases: when A is the infinitesimal generator of an integral semigroup SŽ t .4t G 0 or when K is a partially integral operator. Equation Ž0.1. describes, in the first case, the process of diffusion with jumps w13, 14x and, in the second case, a randomly perturbed dynamical system w11x. In Section 3 we consider two other examples of random movement: a jump process connected with an iterated function system and a randomly controlled dynamical system. In these examples both the semigroup SŽ t .4t G 0 and the operator K are singular Žhave no integral parts. but the semigroup P Ž t .4t G 0 is partially integral.
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1. STABILITY OF CONTINUOUS MARKOV SEMIGROUPS Let Ž X, ⌺, . be a -finite measure space. Denote by D s DŽ X, ⌺, . the subset of L1 s L1 Ž X, ⌺, . which contains all densities, i.e., D s f g L1 : f G 0, 5 f 5 s 1 4 , where 5 ⭈ 5 stands for the norm in L1. A linear mapping P: L1 ª L1 is called a Marko¨ operator if P Ž D . ; D. Let P be a given Markov operator. A density f# is called in¨ ariant if Pf# s f#. The operator P is called asymptotically stable if there is an invariant density f# such that lim 5 P n f y f# 5 s 0
nª⬁
for f g D.
Ž 1.1.
Let f be a density with f ) 0 a.e. Define the set C by
½
Cs xgX :
⬁
Ý
5
P nf Ž x. s ⬁ .
ns0
Ž 1.2.
This definition is independent of the choice of f. The Markov operator P is called conser¨ ati¨ e if C s X. From this definition it follows immediately that if P has an invariant density f# ) 0 a.e., then P is conservative. An operator Q: L1 ª L1 is called a kernel operator if it is of the form Qf Ž x . s q Ž x, y . f Ž y . Ž dy . ,
H
Ž 1.3.
where q, also called a kernel, is a measurable non-negative function. Any Markov operator P can be written in the form P s Q q R, where R is a non-negative contraction on L1, Q is a kernel operator, and there is no kernel K with K F R and K k 0. Fix a Markov operator P and let P n s Q n q R n be the decomposition of P n into kernel and singular parts. The operator P is called a pre-Harris operator if ⬁
HX Ý q Ž x, y . Ž dy . ) 0 n
x y a.e.,
Ž 1.4.
ns1
where qn is the kernel corresponding to Q n . It is convenient to formulate condition Ž1.4. in a different way. For any f g L1 the support of f is defined up to a set of measure zero by the formula supp f s x g X : f Ž x . / 0 4 . Let U be a linear, continuous, and positive operator on the space L1. It is easy to check that if supp f 1 s supp f 2 then supp Uf 1 s supp Uf 2 . This
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allows us to define the operator U: ⌺ ª ⌺ by UA s supp Uf if supp f s A, f G 0, and f g L1. This definition is independent of the choice of the function f assuming that sets equal a.e. represent the same element. Condition Ž1.4. takes the following form:
D Qn X s X .
Ž 1.5.
ng⺞
If P is a pre-Harris operator, P is conservative, and Ž X . s 1, then P is called a Harris operator. If instead of Ž1.4. the operator P satisfies the condition ⬁
HXHX Ý q Ž x, y . Ž dy . Ž dx . ) 0, n
Ž 1.6.
ns1
then the operator P is called partially integral. According to w6, Chap. V, Lemma Bx if P is a Markov operator and Q is an integral operator then the operators PQ and QP are integral operators. From this it follows immediately that P n Q m F Q nqm
and
Q n P m F Q nqm .
Ž 1.7.
A semigroup P Ž t .4t G 0 of linear operators on L1 is said to be a Marko¨ semigroup if P Ž t . is a Markov operator for every t G 0 and if for every f g L1 the function t ¬ P Ž t . f is continuous. A density f# is called invariant under the semigroup P Ž t .4t G 0 if P Ž t . f# s f# for every t G 0. The semigroup P Ž t .4t G 0 is called asymptotically stable if there is an invariant density f# such that lim 5 P Ž t . f y f# 5 s 0
tª⬁
for f g D.
It is easy to show that a semigroup P Ž t .4t G 0 is asymptotically stable if for some t 0 ) 0 the operator P Ž t 0 . is asymptotically stable. A semigroup P Ž t .4t G 0 is called partially integral if for some t 0 ) 0, the operator P Ž t 0 . is partially integral. Let P Ž nt 0 . s QŽ nt 0 . q RŽ nt 0 . be the decomposition of P Ž nt 0 . into kernel and singular parts and let qnŽ t 0 . be the kernel corresponding to QŽ nt 0 .. The main results of this section are the following THEOREM 1. Let Ž X, ⌺, . be a -finite measure space and let P Ž t .4t G 0 be a partially integral Marko¨ semigroup. Assume that the semigroup P Ž t .4t G 0 has an in¨ ariant density f#. Suppose that there does not exist a set E g ⌺ such that Ž E . ) 0, Ž X _ E . ) 0, and P Ž t . E s E for all t ) 0. Then the semigroup P Ž t .4t G 0 is asymptotically stable.
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THEOREM 2. Let Ž X, ⌺, . be a -finite measure space and let P Ž t .4t G 0 be a partially integral Marko¨ semigroup. Assume that the semigroup P Ž t .4t G 0 has the only one in¨ ariant density f#. If f# ) 0 a.e. then the semigroup P Ž t .4t G 0 is asymptotically stable. Remark 1. The assumption in Theorem 1 that the semigroup P Ž t .4t G 0 has an invariant density can be replaced by a weaker one that the semigroup has a non-zero invariant function f g L1. Indeed, we can assume that 5 fq5 ) 0. Since P Ž t . is a Markov operator and P Ž t . f s f, we have P Ž t . fqs fq. Hence the function f# s fqr5 fq5 is an invariant density. Theorem 2 is a simple consequence of Theorem 1. Indeed, if f# is the only one invariant density then there is no non-trivial set E such that P Ž t . E s E. Otherwise the function g s f#1 Er5 f#1 E 5 is another invariant density under the semigroup P Ž t .4t G 0 . We precede the proof of Theorem 1 by the following lemma. LEMMA 1. Let Ž X, ⌺, . be a -finite measure space. Assume that P Ž t .4t G 0 is a partially integral Marko¨ semigroup. Suppose that for e¨ ery t G 0 the operator P Ž t . is conser¨ ati¨ e and P Ž t . X s X. Then for e¨ ery t ) 0 the operator P Ž t . is a pre-Harris operator or there exists a set E g ⌺ such that Ž E . ) 0, Ž X _ E . ) 0 and P Ž t . E s E for all t G 0. Proof. First we check that Q Ž t1 . X ; Q Ž t 2 . X
for t 1 - t 2 .
Ž 1.8.
From Ž1.7. it follows that for every s ) 0 and t ) 0 we have QŽ s. P Ž t . X ; QŽ s q t . X
and
P Ž s . Q Ž t . X ; Q Ž s q t . X . Ž 1.9.
Since P Ž t . X s X, we have QŽ s . P Ž t . X s QŽ s . X, for s, t ) 0. This and Ž1.9. imply that QŽ s . X ; QŽ s q t . X for all s ) 0 and t ) 0, which proves Ž1.8.. From Ž1.8. it follows that
D Q Ž nt1 . X s D Q Ž nt2 . X
ng⺞
for 0 - t 1 , 0 - t 2 .
Ž 1.10.
ng⺞
Fix t ) 0 and define E s D ng ⺞ QŽ nt . X. According to Ž1.10. the definition of the set E is independent of the choice of the number t. Since the semigroup P Ž t .4t G 0 is partially integral, Ž E . ) 0. If E s X then Dng ⺞ QŽ nt . X s X for all t ) 0 and consequently for every t ) 0 the operator P Ž t . is a pre-Harris operator. Now, suppose that Ž X _ E . ) 0. From Ž1.9. we obtain P Ž t . Q Ž ns . X ; Q Ž ns q t . X ; Q Ž n Ž t q s . . X for t ) 0,
s ) 0,
ng⺞
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and consequently PŽ t. E ; E
for all t ) 0.
Ž 1.11.
Suppose that there exists a set E0 ; E with Ž E0 . ) 0 such that P Ž s . E s E _ E0
for some s ) 0.
Ž 1.12.
Then for all f g L1 we have
HX_E P Ž s . f d q HE P Ž s . f d 0
s
HX_Ž E_E . Ž P Ž s . Ž f 1
X_E
. q P Ž s . Ž f 1 E . . d
0
F
HXP Ž s . Ž f 1
X_E
. d s H
X_E
f d .
Using induction argument we obtain n
HX_E P Ž ns . f d q Ý HE P Ž ks . f d F HX_E f d ks1
0
for f g L1. Since the operator P Ž s . is conservative, the sum on the left hand side is infinite and consequently HX _ E f d s ⬁, which is impossible. Thus condition Ž1.12. does not hold and P Ž t . E s E for all t ) 0. Remark 2. If we assume that Ž X, ⌺, . is a finite measure space and P Ž t .4t G 0 is a partially integral Markov semigroup such that P Ž t .1 X s 1 X for all t G 0, then from condition Ž1.11. follows easily that P Ž t .1 E s 1 E for all t ) 0, where 1 E denotes the characteristic function of the set E. In this case we have the following alternative: the operator P Ž t . is a pre-Harris operator for every t G 0 or there exists a set E g ⌺ such that Ž E . ) 0, Ž X _ E . ) 0, and P Ž t .1 E s 1 E for all t ) 0. Proof of Theorem 1. Let us observe that supp f# s X. This follows immediately from the fact that P Žsupp f#. s supp f#. Define a new measure space Ž X, ⌺, . with d s f# d and consider the operators P Ž t . f s Ž 1rf# . P Ž t . Ž f ⭈ f# . ,
t ) 0.
Ž 1.13.
Then P Ž t .4t G 0 is also a partially integral Markov semigroup on the space L1 Ž X, ⌺, .. It is easy to observe that the semigroup P Ž t .4t G 0 is asymptotically stable if and only if the semigroup P Ž t .4t G 0 is asymptotically stable. Thus it is sufficient to show that the semigroup P Ž t .4t G 0 is asymptotically
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stable. From definition of the measure it follows that Ž X . s 1. From Ž1.13. we have P Ž t .1 X s 1 X , t ) 0 and consequently the operator P Ž t ., t ) 0 is conservative and P Ž t . X s X, t ) 0. According to Lemma 1 the operator P Ž t . is a Harris operator for every t G 0. Fix t 0 ) 0 and denote P s P Ž t 0 .. It is easy to check that the semigroup P Ž t .4t G 0 is asymptotically stable if and only if P n f ª Hf d for f g L1 Ž X, ⌺, .. In order to check asymptotic stability of the operator P we use some properties of Harris operators w6x. Denote ⌺ d s A g ⌺ : P n 1 A s 1 B n , for n s 1, 2, . . . 4 .
Ž 1.14.
Then ⌺ d is an atomic -algebra. We claim that ⌺ d s X, ⭋4 . In order to show this fact we introduce an auxiliary function
A Ž t . s Ž P Ž t . A .
for t G 0,
A g ⌺.
Ž 1.15.
Let 0 - s - t. Since the operator P Ž t y s . is a Markov operator and P Ž t .1 A F 1 P Ž t . A , we have
A Ž s . s
HX1
P Ž s. A
d s
HXP Ž t y s . 1
P Ž s. A
d F Ž P Ž t . A . s A Ž t . .
Ž 1.16. This implies that AŽ t . is a non-decreasing function of t. Let A g ⌺ d and P n 1 A s 1 B n. Since P is a Markov operator we have Ž Bn . s Ž A. and AŽ nt 0 . s AŽ0. for n G 0. Consequently AŽ t . s AŽ0. for t G 0 and A g ⌺ d . Since
H1
A
d s P Ž t . 1 A d F 1 P Ž t . A d s 1 A d
H
H
H
we have P Ž t .1 A s 1 P Ž t . A for t G 0 and A g ⌺ d . This implies that P Ž t . A g ⌺ d for t G 0 and A g ⌺ d . Now let E g ⌺ d be an atom and let ␦ ) 0 be such a number that 5 P Ž s. 1E y 1E 5 - Ž E.
Ž 1.17.
for s F ␦ . From P Ž t .1 E s 1 P Ž t . E and Ž1.17. it follows that P Ž s . E l E / ⭋. Since P Ž s . E g ⌺ d and E is an atom, we have P Ž s . E s E for s F ␦ . Consequently P Ž t . E s E for t G 0 and E g ⌺ d . This implies that ⌺ d s X, ⭋4 . Since ⌺ d is a trivial -algebra, from w6, formula Ž8.7.x it follows that P n f ª Hf d for f g L1 Ž X, ⌺, . which completes the proof. Remark 3. The assumption in Theorems 1 and 2 that the semigroup P Ž t .4t G 0 is a partially integral is essential. Let X s S 1 be a unit circle on
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675
the complex plain with centre z 0 s 0, let ⌺ s BŽ X . be the -algebra of Borel subsets of X, and let be the arc-Lebesgue measure on X. Then the Markov semigroup P Ž t .4t G 0 given by P Ž t . f Ž z . s f Ž ze i t . has only one invariant density, but it is not asymptotically stable.
2. TRANSPORT EQUATIONS Now we give some examples of partial differential equations which generate Markov semigroups. We also give some applications of Theorem 1 to these equations. In this section X s ⺢ d, ⌺ s BŽ X . is the -algebra of Borel subsets of X and is the Lebesgue measure on X. We write d s dx. 2.1. Fokker᎐Planck Equation In the space X the Fokker᎐Planck equation has the form
⭸u ⭸t
d
s
Ý i , js1
⭸ 2 Ž i j Ž x . u . ⭸ xi ⭸ x j
d
y
Ý is1
⭸ Ž ai Ž x . u . ⭸ xi
,
u Ž x, 0 . s ¨ Ž x . . Ž 2.1.
We assume that the functions i j and a i are bounded and sufficiently smooth. We also assume that d
Ý
i j Ž x . i j G ␣ < < 2
i , js1
for some ␣ ) 0 and every g ⺢ d and x g X. The solution of this equation describes the distribution of a diffusion process. This equation generates a Markov semigroup given by P Ž t . ¨ Ž x . s uŽ x, t ., where ¨ Ž x . s uŽ x, 0.. The Markov semigroup generated by the Fokker᎐Planck equation is an integral semigroup. That is, PŽ t. f Ž x. s
HXq Ž t , x, y . f Ž y . dy,
t ) 0.
Since the kernel q is positive we have P Ž t . E s X for t ) 0 and every measurable set E with positive Lebesgue measure. This implies that E s X is the only invariant set with respect to the semigroup P Ž t .4t G 0 . Thus this semigroup is asymptotically stable if and only if it has an invariant density f# Ži.e., Af# s 0, where A is the infinitesimal generator of the semigroup P Ž t .4t G 0 ..
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If we assume that i j ' 0 in Eq. Ž2.1., then we obtain the Liou¨ ille equation d ⭸u ⭸ sy Ý Ž 2.2. Ž ai Ž x . u . . ⭸t is1 ⭸ x i As in the previous example, Eq. Ž2.2. generates a Markov semigroup given by P Ž t . ¨ Ž x . s uŽ x, t ., where ¨ Ž x . s uŽ x, 0.. The semigroup P Ž t .4t G 0 can be given explicitly. Namely, for each x g X denote by t x the solution x Ž t . of the equation x⬘ Ž t . s a Ž x Ž t . .
Ž 2.3.
with the initial condition x Ž0. s x. Then P Ž t . f Ž x . s f Ž yt x . det
d dx
yt x
for f g L1 Ž X . .
Ž 2.4.
Equation Ž2.2. has the following interpretation. In the space X we consider the movement of points given by Eq. Ž2.3.. We look at this movement statistically; that is, we consider the evolution of densities of the distribution of points. Then this evolution is described by Eq. Ž2.2.. The semigroup generated by Eq. Ž2.2. is not partially integral and not asymptotically stable but, as we show in next examples, stochastic perturbations of this semigroup can lead to asymptotically stable semigroups. 2.3. Transport Equations If the equation ⭸⭸ut s Au generates a Markov semigroup S Ž t .4t G 0 , K is a Markov operator, and ) 0, then the equation
⭸u ⭸t
s Au y u q Ku
Ž 2.5.
also generates a Markov semigroup. Let A f s Af y f q Kf. Then Eq. Ž2.5. can be rewritten as the evolution equation u⬘ Ž t . s A u, Ž 2.6. where the solution u is a function from w0, ⬁. to L1 Ž X . and u satisfies the initial condition uŽ0. s f, f g L1 Ž X .. From the Phillips perturbation theorem w5x, Eq. Ž2.6. generates a continuous semigroup of Markov operators P Ž t .4t G 0 on L1 Ž X . given by P Ž t . f s u Ž t . s ey t
⬁
Ý nSn Ž t . f , ns0
Ž 2.7.
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677
where S0 Ž t . s SŽ t . and Snq 1 Ž t . f s
t
H0 S Ž t y s . KS Ž s . f ds, n
n G 0.
Ž 2.8.
The semigroup P Ž t .4t G 0 satisfies the integral equation P Ž t . f s ey t S Ž t . f q
t y s
H0 e
S Ž s . KP Ž t y s . f ds.
Ž 2.9.
If the semigroup S Ž t .4t G 0 is partially integral or the operator K is partially integral then from Ž2.9. it follows that the semigroup P Ž t .4t G 0 is partially integral. From Ž2.9. and continuity of the semigroups SŽ t .4t G 0 and P Ž t .4t G 0 it follows that for a measurable set E we have P Ž t . E ; E for all t G 0 if and only if KE ; E and SŽ t . E ; E for all t G 0. In particular, if A is the operator from Eq. Ž2.1. then P Ž t .4t G 0 is asymptotically stable if and only if it has an invariant density f# Ži.e., A f# s 0.. In many applications A is the operator from Eq. Ž2.2. and the Markov operator K corresponds to some transition probability function P Ž x, E .; i.e., K *1 E Ž x . s P Ž x, E . for x g X and E g ⌺, where K * denotes the adjoint operator of K. We recall that P : X = ⌺ ª w0, 1x is a transition probability function if for each x g X the function mŽ E . s P Ž x, E . is a probabilistic measure and for each E g ⌺ the function f Ž x . s P Ž x, E . is ⌺-measurable. In this case Eq. Ž2.5. has an interesting probabilistic interpretation. Consider a collection of particles moving under the action of the equation x⬘ s aŽ x .. This motion is modified in the following way. In every time interval w t, t q ⌬ t x a particle with the probability P Ž x, E . ⌬ t q oŽ ⌬ t . changes its position from x to a point from the set E. Then any solution of Ž2.5. is the probability density function of the position of the particle at time t. Note that if the operator K is given by transition probability function P then KE ; E iff P Ž x, E . s 1 for a.e. x g E. In the next section we consider two examples of random movement of this type. In these examples both the semigroup SŽ t .4t G 0 and the operator K are singular Žhave no integral parts. but the semigroup P Ž t .4t G 0 is partially integral. Moreover we give sufficient conditions for asymptotic stability of these semigroups.
3. FURTHER APPLICATIONS In this section X is an open subset of ⺢ d, BŽ X . is the -algebra of Borel subsets of X, and is the Lebesgue measure on X.
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Now we consider a jump process connected with an iterated function system. Let Ti : X ª X, for i s 1, . . . , k, be a sequence of continuously differentiable transformations. Let pi : X ª w0, 1x, i s 1, . . . , k, be a sequence of continuous functions such that Ý kis1 pi Ž x . s 1 for each x g X. As before we consider particles which move along the solutions of the equation x⬘ s aŽ x .. At any time interval w t, t q ⌬ t x a particle with the probability pi Ž x . ⌬ t q oŽ ⌬ t . jumps from the point x to Ti Ž x .. Assume that det TiX Ž x . / 0 for almost every x. Then the transformations Ti are nonsingular; that is, if Ž E . s 0 then ŽTiy1 Ž E .. s 0. Let K T i : L1 Ž X . ª L1 Ž X . be the Frobenius᎐Perron operator corresponding to the transformation Ti , i.e., the operator satisfying the condition
HE K
Ti
f Ž x . dx s
HT
y1 i Ž E.
f Ž x . dx
for every measurable set E and f g L1 Ž X, BŽ X ., .. Then Eq. Ž2.5. with the operator K given by the formula k
Kf Ž x . s
Ý K T Ž f Ž x . pi Ž x . .
Ž 3.1.
i
is1
describes the evolution of the densities of the distribution of the above jump process. Assume that t Ž X . ; X for all t G 0. Then Eq. Ž2.2. generates a semigroup SŽ t .4t G 0 of Markov operators on the space L1 Ž X, BŽ X ., ., given by S Ž t . f Ž x . s 1 X Ž yt x . f Ž yt x . det
d dx
yt x ,
Ž 3.2.
where t x denote the solution x Ž t . of the equation x⬘ Ž t . s a Ž x Ž t . . with the initial condition x Ž0. s x. Let P Ž t .4t G 0 be the semigroup generated by Eq. Ž2.5. and let E be a measurable set. We have P Ž t . E ; E for all t G 0 if and only if Ti Ž E . ; E for all i s 1, . . . , k and t Ž E . ; E for all t G 0. PROPOSITION 1. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. Let Ž i1 , . . . , i d . be a gi¨ en sequence of integers from the set 1, . . . , k 4 . Let x 0 g X be a gi¨ en point and let x j s Ti jŽ x jy1 . for j s 1, . . . , d. Set X
X
¨ j s Ti dŽ x dy1 . ⭈⭈⭈ Ti j Ž x jy1 . a Ž x jy1 . y a Ž x d .
679
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
for j s 1, . . . , d. Assume that pi jŽ x jy1 . ) 0 for all j s 1, . . . , d and suppose that the ¨ ectors ¨ 1 , . . . , ¨ d are linearly independent. Then the semigroup P Ž t .4t G 0 is asymptotically stable. Proof. According to Theorem 1 and Remark 1, it is sufficient to prove that the semigroup P Ž t .4t G 0 is partially integral. We check that for some t ) 0, the operator P Ž t . is partially integral. Precisely, we show that there exist t ) 0, ) 0, and neighbourhoods U of x 0 and V of x d such that the operator P Ž t . is partially integral with the kernel k Ž x, y . G 1 V Ž x .1 U Ž y .. Let y g X and t ) 0. Let ⌬ t s s Ž 1 , . . . , d . : i ) 0, 1 q ⭈⭈⭈ qd F t 4 . Define a function y, t on the set ⌬ t by
y , t Ž . s ty 1y 2y ⭈⭈⭈ y d (Ti d ( d ( ⭈⭈⭈ (Ti 2 ( 2 (Ti1 ( 1Ž y . . Let M be the matrix with columns ¨ 1 , . . . , ¨ d . It is easy to check that lim lim
dy , t Ž . d
yªx 0 tª0
s M.
Ž 3.3.
Since det M / 0, we have det
dy , t Ž . d
/0
Ž 3.4.
for y sufficiently close to x 0 and t sufficiently small. Let S d Ž t . be given by Ž2.8.. Then from Ž2.7. it follows that P Ž t . f G ey t dS d Ž t . f for f G 0. Let K Ž t, . be the operator given by K Ž t , . s S Ž t y 1 y 2 y ⭈⭈⭈ yd . KS Ž d . ⭈⭈⭈ KS Ž 1 . and let K *Ž t, . be the adjoint operator of K Ž t, .. Then for every measurable set E we have y t d
HEP Ž t . f Ž x . dx G e
s ey t d
H⌬ HE K Ž t , . f Ž y . dy d t
H⌬ HX f Ž y . K * Ž t , . 1
E
Ž y . dy d .
Ž 3.5.
t
We have K * Ž t , . s S* Ž 1 . K * ⭈⭈⭈ S* Ž d . K *S* Ž t y 1 y 2 y ⭈⭈⭈ yd . , where S*Ž . and K * are the adjoint operators of SŽ . and K. Since S*Ž . f Ž y . s f Ž y ., K *f Ž y . s Ý kis1 pi Ž y . f ŽTi Ž y .., and pi jŽ x jy1 . ) 0, there
PICHOR ´ AND RUDNICKI
680
exists a constant c1 ) 0 such that K * Ž t , . 1 E Ž y . G c1 1 E Ž y , t Ž . .
Ž 3.6.
for y sufficiently close to x 0 and t sufficiently small. From Ž3.5. and Ž3.6. it follows that there exist a constant c 2 and a neighbourhood U of x 0 such that
HEP Ž t . f Ž x . dx G c HU f Ž y . H⌬ 1 Ž E
2
y, t
Ž . . d dy.
Ž 3.7.
t
Substituting x s y, t Ž . to Ž3.7. and using Ž3.4. we obtain
HE P Ž t . f Ž x . dx G HU f Ž y . H
1 E Ž x . dx dy,
y, t Ž ⌬ t .
Ž 3.8.
where is a positive constant. Now decreasing t and the neighbourhood U of x 0 we find a neighbourhood V of x d such that V ; y, t Ž ⌬ t . for y g U. This and Ž3.8. imply that PŽ t. f Ž x. G
HX1
V
Ž x . 1U Ž y . f Ž y . dy,
which completes the proof. Now, consider the case when X is an open subset of ⺢ and we have only one transformation T : X ª X. We assume that T is continuously differentiable and T ⬘Ž x . / 0 for almost every x. Now K is the Frobenius᎐Perron operator corresponding to the transformation T. Equation Ž2.5. takes the form
⭸u ⭸t
q u s y
⭸ Ž au . ⭸x
q Ku.
Ž 3.9.
Assume that t Ž X . ; X for all t G 0. Let SŽ t .4t G 0 and P Ž t .4t G 0 be the semigroups generated by Eqs. Ž2.2. and Ž2.5., respectively. In this case we can state Proposition 1 in the following way. COROLLARY 1. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. If aŽT Ž x .. / T ⬘Ž x . aŽ x . for some x g X, then the semigroup P Ž t .4t G 0 is asymptotically stable. EXAMPLE 1. Consider the equation
⭸u ⭸t
Ž x, t . q
⭸u ⭸x
Ž x, t . q u Ž x, t . s 2 u Ž 2 x, t . ,
x ) 0,
t G 0,
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
681
with the boundary and initial conditions: uŽ0, t . s 0 and uŽ x, 0. s f Ž x .. This equation is of the form Ž3.9. with X s Ž0, ⬁., a ' 1, s 1, Kf Ž x . s 2 f Ž2 x ., and T Ž x . s 2x . We show that the Markov semigroup P Ž t .4t G 0 generated by this equation is asymptotically stable. First we check that X is the only invariant set with respect to the semigroup P Ž t .4t G 0 . Indeed, assume that P Ž t . E ; E for some measurable set E and all t G 0. Then SŽ t . E ; E for all t G 0 and T Ž E . ; E. Since SŽ t . f Ž x . s f Ž x y t .1 w t, ⬁.Ž x ., we have E s Ž ␣ , ⬁. for some ␣ G 0. Now the condition T Ž E . ; E implies that E s X. Then the function f# given by f# Ž x . s
⬁
Ý an ey2
n
x
,
ns0
where n
a0 s 1
and
Ž y2. an s 2 Ž 2 y 1 . Ž 2 y 1 . ⭈⭈⭈ Ž 2 n y 1 .
for n G 1,
is invariant. Since T ⬘Ž x . / 1, according to Corollary 1 the semigroup P Ž t .4t G 0 is asymptotically stable. 3.2. Randomly Controlled Dynamical System Let Yt be a continuous time Markov chain on the phase space ⌫ s 1, . . . , k 4 , k G 2, such that the transition probability from the state j to the state i / j in time interval ⌬ t equals pi j ⌬ t q oŽ ⌬ t .. We assume that pi j ) 0 for all i / j. Let a be a d-dimensional vector function defined on X = ⌫. Let 0 be a d-dimensional random variable independent of Yt . Consider the stochastic differential equation d t s a Ž t , Yt . dt. The pair Ž t , Yt . constitutes a Markov process on X = ⌫. The process t describes the movement of points under the action of k dynamical systems x⬘ s aŽ x, i ., i s 1, . . . , k. The Markov chain Yt decides which dynamical systems acts at time t. We assume that the random variable 0 has an absolutely continuous distribution. Then the random variable t has an absolutely continuous distribution for each t ) 0. Define the function u by the formula Prob Ž Ž t , Yt . g E = i 4 . s
HEu Ž x, i , t . dx.
We assume that a j Ž⭈, i . are continuously differentiable and bounded functions for all i, j. Denote by A i the differential operators d
Ai f s y
Ý js1
⭸ Ž a j Ž x, i . f . ⭸ xj
.
Ž 3.10.
682
PICHOR ´ AND RUDNICKI
Let pii s yÝ j/ i pji and denote by M the matrix w pi j x. We use the notation u i Ž x, t . s uŽ x, i, t . and u s Ž u1 , . . . , u k . is a vertical vector. Then the vector u satisfies the equation
⭸u ⭸t
s Mu q Au,
Ž 3.11.
where Au s Ž A1 u1 , . . . , A k u k . is also a vertical vector. Assume that for each 1 F i F k and for all t G 0 we have ti Ž X . ; X. Then the operator A i generates a semigroup SŽ t .Ž i .4t G 0 of Markov operators on the space L1 Ž X, BŽ X ., .. Let BŽ X = ⌫ . be the -algebra of Borel subsets of X = ⌫ and let m be the product measure on BŽ X = ⌫ . given by mŽ B = i4. s Ž B . for each B g BŽ X . and 1 F i F k. The operator A generates a continuous semigroup SŽ t .4t G 0 of Markov operators on the space L1 Ž X = ⌫, BŽ X = ⌫ ., m. given by the formula S Ž t . f s Ž S Ž t . Ž 1. f 1 , . . . , S Ž t . Ž k . f k . , where f i Ž x . s f Ž x, i . for x g X, 1 F i F k. Now, let be a constant such that s max yp11 , . . . , ypk k 4 and K s y1 M q I. Then Eq. Ž3.11. can be written in the form
⭸u ⭸t
s Au y u q Ku
Ž 3.12.
and the matrix K is a Markov operator on L1 Ž X = ⌫, BŽ X = ⌫ ., m.. Let SŽ t .4t G 0 and P Ž t .4t G 0 be the semigroups generated by the equation ⭸u Ž . ⭸ t s Au and Eq. 3.12 , respectively. A measurable set E ; X = ⌫ is invariant with respect to the semigroup P Ž t .4t G 0 if and only if E s E0 = ⌫ and
ti Ž E0 . s E0
for t G 0 and
i s 1, . . . , k.
Ž 3.13.
Let Ž i1 , . . . , i dq1 . be a sequence of integers from the set 1, . . . , k 4 . For x g X and t ) 0 we define the function x, t on the set ⌬ t by i dq 1 id i2 i1 x , t Ž 1 , . . . , d . s ty 1 y 2 y ⭈⭈⭈ y d ( d ( ⭈⭈⭈ ( 2 ( 1 Ž x . .
PROPOSITION 2. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. Suppose that for some x 0 g X, t 0 ) 0, and 0 g ⌬ t 0 we ha¨ e det
d x 0 , t 0Ž 0 . d
/ 0.
Then the semigroup P Ž t .4t G 0 is asymptotically stable.
Ž 3.14.
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683
Proof. The proof is similar to the proof of Proposition 1 and we only sketch it. Since the elements of the matrix K are positive constants, there exists a constant c such that
HEP Ž t . f Ž x, i
dq 1
. dx G cH f Ž y, i1 . H 1 E Ž y , t Ž . . d dy ⌬t
X
Ž 3.15.
for every measurable set E and all t from some neighbourhood of t 0 . From Ž3.14. and Ž3.15. we conclude that there exist ) 0 and neighbourhoods U of x 0 and V of x 0 , t 0Ž 0 . such that P Ž t . f Ž x, i dq 1 . G
HX1
V
Ž x . 1U Ž y . f Ž y, i1 . dy,
which proves that the semigroup P Ž t .4t G 0 is partially integral. Consider the case k s d q 1. Set ¨ j Ž x . s aŽ x, j . y aŽ x, d q 1. for j s 1, . . . , d and x g X. If we assume that for some point x 0 g X the vectors ¨ 1Ž x 0 ., . . . , ¨ d Ž x 0 . are linearly independent then condition Ž3.14. holds for t 0 sufficiently small. Indeed, denote by W the matrix with columns ¨ 1Ž x 0 ., . . . , ¨ d Ž x 0 .. Then det W / 0. Moreover, one can check that lim lim
xªx 0 tª0
dx , t Ž . d
s W,
which implies that there is a sufficiently small t 0 such that condition Ž3.14. holds. Now, assume that X is an open subset of ⺢ and ⌫ s 1, 24 . Consider the differential operators A j f Ž x . s y dxd Ž aŽ x, j . f Ž x .., j s 1, 2 and assume that tj Ž X . ; X for j s 1, 2 and for all t ) 0. Moreover assume that p12 ) 0 and p 21 ) 0. Suppose that for some point x 0 g X we have aŽ x 0 , 1. / aŽ x 0 , 2.. Then there is a sufficiently small t 0 such that condition Ž3.14. holds. If P Ž t .4t G 0 is the semigroup generated by Eq. Ž3.12. then according to Proposition 2 we have the following corollary. COROLLARY 2. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. Moreo¨ er suppose that for some point x 0 g X we ha¨ e aŽ x 0 , 1. / aŽ x 0 , 2.. Then the semigroup P Ž t .4t G 0 is asymptotically stable. EXAMPLE 2. Consider the following system of equations:
⭸ u1 ⭸t ⭸ u2 ⭸t
s ypu1 q pu 2 y s pu1 y pu 2 y
⭸ Ž a Ž x, 1 . u1 . ⭸x
⭸ Ž a Ž x, 2 . u 2 . ⭸x
,
Ž 3.16. .
PICHOR ´ AND RUDNICKI
684
Here p ) 0 and the functions aŽ x, 1. and aŽ x, 2. are continuously differentiable and bounded. We assume that there are some constants c ) 0, L ) 0 such that aŽ x, 1. G c, aŽ x, 2. F yc for all x g ⺢ and
Ž sign x .
1 a Ž x, 1 .
q
1 a Ž x, 2 .
for < x < G L,
Gc
Ž 3.17.
where sign x s xr< x <. Then the semigroup P Ž t .4t G 0 generated by Ž3.16. is asymptotically stable. Indeed, from Ž3.13. it follows immediately that the set ⺢ = 1, 24 is the only invariant set with respect to the semigroup P Ž t .4t G 0 . Let
½
b Ž x . s exp yp
H0
x
1 a Ž s, 1 .
q
1 a Ž s, 2 .
5
ds .
From Ž3.17. it follows that the function f#Ž x, i . s < a bŽ Žx,x i.. < is integrable. Since f# is also a stationary solution of Ž3.16. and aŽ x, 1. ) 0 ) aŽ x, 2., the semigroup P Ž t .4t G 0 is asymptotically stable. REFERENCES 1. K. Baron and A. Lasota, Asymptotic properties of Markov operators defined by Volterra type integrals, Ann. Polon. Math. 58 Ž1993., 161᎐175. 2. W. Bartoszek and T. Brown, On Frobenius᎐Perron operators which overlap supports, Bull. Polish Acad. Math. 45 Ž1997., 17᎐24. 3. S. Chandrasekhar and G. Munch, The theory of fluctuations in brightness of the ¨ Milky-Way, Astrophys. J. 125, 94᎐123. 4. O. Diekmann, H. J. A. Heijmans, and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 Ž1984., 227᎐248. 5. N. Dunford and J. T. Schwartz, ‘‘Linear Operators, Part I,’’ Interscience, New York, 1968. 6. S. R. Foguel, ‘‘The Ergodic Theory of Markov Processes,’’ Van Nostrand Reinhold, New York, 1969. 7. A. Lasota and M. C. Mackey, ‘‘Chaos, Fractals and Noise. Stochastic Aspects of Dynamics,’’ Springer Applied Mathematical Sciences, Vol. 97, Springer-Verlag, New York, 1994. 8. J. Łuczka and R. Rudnicki, Randomly flashing diffusion: asymptotic properties, J. Statist. Phys. 83 Ž1996., 1149᎐1164. 9. M. C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 Ž1994., 89᎐109. 10. J. Malczak, An application of Markov operators in differential and integral equations, Rend. Sem. Mat. Uni¨ . Pado¨ a 87 Ž1992., 281᎐297. 11. K. Pichor, ´ Asymptotic stability of a partial differential equation with an integral perturbation, Ann. Polon. Math. 68 Ž1998., 83᎐96. 12. K. Pichor ´ and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl. 215 Ž1997., 56᎐74.
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13. K. Pichor ´ and R. Rudnicki, Asymptotic behaviour of Markov semigroups and applications to transport equations, Bull. Polish Acad. Math. 45 Ž1997., 379᎐397. 14. R. Rudnicki, Asymptotic behaviour of an integro-parabolic equation, Bull. Polish Acad. Math. 40 Ž1992., 111᎐128. 15. R. Rudnicki, Asymptotic behaviour of a transport equation, Ann. Polon. Math. 57 Ž1992., 45᎐55. 16. R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Math. 43 Ž1995., 245᎐262. 17. J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Polish Acad. Math. 44 Ž1996., 230᎐252.
Continuous Markov Semigroups and Stability of Transport Equations 1 Katarzyna Pichor ´ Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland
and Ryszard Rudnicki Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland; and Institute of Mathematics, Silesian Uni¨ ersity, Bankowa 14, 40-007 Katowice, Poland Submitted by Paul S. Muhly Received November 22, 1999
A new theorem for asymptotic stability of Markov semigroups is proved. This result is applied to transport equations connected with diffusion and jumping processes and randomly perturbed dynamical systems. 䊚 2000 Academic Press Key Words: Markov semigroup; asymptotic stability; transport equation; Markov process.
0. INTRODUCTION Markov operators and Markov semigroups are intensively studied because they play a special role in applications. The book of Lasota and Mackey w7x is an excellent survey of many results on this subject. The main problem of the theory of Markov operators is their asymptotic stability, i.e., when there exists an invariant density f# such that P n f ª f# in L1 for 1 This research was supported by the State Committee for Scientific Research ŽPoland. Grant 2 P03A 010 16 and by the Foundation for Polish Science. Part of this work was done during a visit at the University of Pau. The authors thank French᎐Polish Integrated Action Programme Polonium for financial support of this visit.
668 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
669
every density f. This problem has been recently investigated for Markov operators with a nontrivial integral part w1, 2, 10, 16x. In particular in w16x was shown that if such an operator has a positive invariant density f# and has no other periodic points in the set of densities, then it is asymptotically stable. The purpose of this paper is to provide new sufficient conditions for asymptotic stability of Markov semigroups and apply them to partial differential equations. In Section 1 we strengthen the results of w16x in the case of continuous time Markov semigroups. Namely, we show that if a partially integral Markov semigroup has only one invariant density f# and f# ) 0 a.e., then it is asymptotically stable. The proof of this theorem is based on the results concerning properties of Harris operators w6x. The question of asymptotic stability of positive semigroups can be investigated also by means of methods of spectral analysis. Such results are based on the assumption that there exists a strictly dominant real eigenvalue of the generator of the semigroup. Verification of this assumption requires the analysis of the whole spectrum of the generator, which is technically difficult. Using our result it is sufficient to check that the semigroup has only one invariant density. Moreover, we do not need to prove that the semigroup is eventually compact. In Sections 2 and 3 we apply results concerning asymptotic stability of Markov semigroups to transport equations. We investigate Markov semigroups generated by the equations
⭸u ⭸t
q u s Au q Ku,
Ž 0.1.
where K is a Markov operator, ) 0, and A is a differential operator of the first or second order. Equation Ž0.1. appears in such diverse areas as astrophysics Žfluctuations in the brightness of the Milky-Way w3x., population dynamics w4, 9x, the theory of jump processes w15, 17x, and multistate diffusion w8, 12x. In order to apply our results we have to check that the semigroup P Ž t .4t G 0 generated by Eq. Ž0.1. is partially integral. In Section 2 we consider two such cases: when A is the infinitesimal generator of an integral semigroup SŽ t .4t G 0 or when K is a partially integral operator. Equation Ž0.1. describes, in the first case, the process of diffusion with jumps w13, 14x and, in the second case, a randomly perturbed dynamical system w11x. In Section 3 we consider two other examples of random movement: a jump process connected with an iterated function system and a randomly controlled dynamical system. In these examples both the semigroup SŽ t .4t G 0 and the operator K are singular Žhave no integral parts. but the semigroup P Ž t .4t G 0 is partially integral.
PICHOR ´ AND RUDNICKI
670
1. STABILITY OF CONTINUOUS MARKOV SEMIGROUPS Let Ž X, ⌺, . be a -finite measure space. Denote by D s DŽ X, ⌺, . the subset of L1 s L1 Ž X, ⌺, . which contains all densities, i.e., D s f g L1 : f G 0, 5 f 5 s 1 4 , where 5 ⭈ 5 stands for the norm in L1. A linear mapping P: L1 ª L1 is called a Marko¨ operator if P Ž D . ; D. Let P be a given Markov operator. A density f# is called in¨ ariant if Pf# s f#. The operator P is called asymptotically stable if there is an invariant density f# such that lim 5 P n f y f# 5 s 0
nª⬁
for f g D.
Ž 1.1.
Let f be a density with f ) 0 a.e. Define the set C by
½
Cs xgX :
⬁
Ý
5
P nf Ž x. s ⬁ .
ns0
Ž 1.2.
This definition is independent of the choice of f. The Markov operator P is called conser¨ ati¨ e if C s X. From this definition it follows immediately that if P has an invariant density f# ) 0 a.e., then P is conservative. An operator Q: L1 ª L1 is called a kernel operator if it is of the form Qf Ž x . s q Ž x, y . f Ž y . Ž dy . ,
H
Ž 1.3.
where q, also called a kernel, is a measurable non-negative function. Any Markov operator P can be written in the form P s Q q R, where R is a non-negative contraction on L1, Q is a kernel operator, and there is no kernel K with K F R and K k 0. Fix a Markov operator P and let P n s Q n q R n be the decomposition of P n into kernel and singular parts. The operator P is called a pre-Harris operator if ⬁
HX Ý q Ž x, y . Ž dy . ) 0 n
x y a.e.,
Ž 1.4.
ns1
where qn is the kernel corresponding to Q n . It is convenient to formulate condition Ž1.4. in a different way. For any f g L1 the support of f is defined up to a set of measure zero by the formula supp f s x g X : f Ž x . / 0 4 . Let U be a linear, continuous, and positive operator on the space L1. It is easy to check that if supp f 1 s supp f 2 then supp Uf 1 s supp Uf 2 . This
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
671
allows us to define the operator U: ⌺ ª ⌺ by UA s supp Uf if supp f s A, f G 0, and f g L1. This definition is independent of the choice of the function f assuming that sets equal a.e. represent the same element. Condition Ž1.4. takes the following form:
D Qn X s X .
Ž 1.5.
ng⺞
If P is a pre-Harris operator, P is conservative, and Ž X . s 1, then P is called a Harris operator. If instead of Ž1.4. the operator P satisfies the condition ⬁
HXHX Ý q Ž x, y . Ž dy . Ž dx . ) 0, n
Ž 1.6.
ns1
then the operator P is called partially integral. According to w6, Chap. V, Lemma Bx if P is a Markov operator and Q is an integral operator then the operators PQ and QP are integral operators. From this it follows immediately that P n Q m F Q nqm
and
Q n P m F Q nqm .
Ž 1.7.
A semigroup P Ž t .4t G 0 of linear operators on L1 is said to be a Marko¨ semigroup if P Ž t . is a Markov operator for every t G 0 and if for every f g L1 the function t ¬ P Ž t . f is continuous. A density f# is called invariant under the semigroup P Ž t .4t G 0 if P Ž t . f# s f# for every t G 0. The semigroup P Ž t .4t G 0 is called asymptotically stable if there is an invariant density f# such that lim 5 P Ž t . f y f# 5 s 0
tª⬁
for f g D.
It is easy to show that a semigroup P Ž t .4t G 0 is asymptotically stable if for some t 0 ) 0 the operator P Ž t 0 . is asymptotically stable. A semigroup P Ž t .4t G 0 is called partially integral if for some t 0 ) 0, the operator P Ž t 0 . is partially integral. Let P Ž nt 0 . s QŽ nt 0 . q RŽ nt 0 . be the decomposition of P Ž nt 0 . into kernel and singular parts and let qnŽ t 0 . be the kernel corresponding to QŽ nt 0 .. The main results of this section are the following THEOREM 1. Let Ž X, ⌺, . be a -finite measure space and let P Ž t .4t G 0 be a partially integral Marko¨ semigroup. Assume that the semigroup P Ž t .4t G 0 has an in¨ ariant density f#. Suppose that there does not exist a set E g ⌺ such that Ž E . ) 0, Ž X _ E . ) 0, and P Ž t . E s E for all t ) 0. Then the semigroup P Ž t .4t G 0 is asymptotically stable.
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672
THEOREM 2. Let Ž X, ⌺, . be a -finite measure space and let P Ž t .4t G 0 be a partially integral Marko¨ semigroup. Assume that the semigroup P Ž t .4t G 0 has the only one in¨ ariant density f#. If f# ) 0 a.e. then the semigroup P Ž t .4t G 0 is asymptotically stable. Remark 1. The assumption in Theorem 1 that the semigroup P Ž t .4t G 0 has an invariant density can be replaced by a weaker one that the semigroup has a non-zero invariant function f g L1. Indeed, we can assume that 5 fq5 ) 0. Since P Ž t . is a Markov operator and P Ž t . f s f, we have P Ž t . fqs fq. Hence the function f# s fqr5 fq5 is an invariant density. Theorem 2 is a simple consequence of Theorem 1. Indeed, if f# is the only one invariant density then there is no non-trivial set E such that P Ž t . E s E. Otherwise the function g s f#1 Er5 f#1 E 5 is another invariant density under the semigroup P Ž t .4t G 0 . We precede the proof of Theorem 1 by the following lemma. LEMMA 1. Let Ž X, ⌺, . be a -finite measure space. Assume that P Ž t .4t G 0 is a partially integral Marko¨ semigroup. Suppose that for e¨ ery t G 0 the operator P Ž t . is conser¨ ati¨ e and P Ž t . X s X. Then for e¨ ery t ) 0 the operator P Ž t . is a pre-Harris operator or there exists a set E g ⌺ such that Ž E . ) 0, Ž X _ E . ) 0 and P Ž t . E s E for all t G 0. Proof. First we check that Q Ž t1 . X ; Q Ž t 2 . X
for t 1 - t 2 .
Ž 1.8.
From Ž1.7. it follows that for every s ) 0 and t ) 0 we have QŽ s. P Ž t . X ; QŽ s q t . X
and
P Ž s . Q Ž t . X ; Q Ž s q t . X . Ž 1.9.
Since P Ž t . X s X, we have QŽ s . P Ž t . X s QŽ s . X, for s, t ) 0. This and Ž1.9. imply that QŽ s . X ; QŽ s q t . X for all s ) 0 and t ) 0, which proves Ž1.8.. From Ž1.8. it follows that
D Q Ž nt1 . X s D Q Ž nt2 . X
ng⺞
for 0 - t 1 , 0 - t 2 .
Ž 1.10.
ng⺞
Fix t ) 0 and define E s D ng ⺞ QŽ nt . X. According to Ž1.10. the definition of the set E is independent of the choice of the number t. Since the semigroup P Ž t .4t G 0 is partially integral, Ž E . ) 0. If E s X then Dng ⺞ QŽ nt . X s X for all t ) 0 and consequently for every t ) 0 the operator P Ž t . is a pre-Harris operator. Now, suppose that Ž X _ E . ) 0. From Ž1.9. we obtain P Ž t . Q Ž ns . X ; Q Ž ns q t . X ; Q Ž n Ž t q s . . X for t ) 0,
s ) 0,
ng⺞
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
673
and consequently PŽ t. E ; E
for all t ) 0.
Ž 1.11.
Suppose that there exists a set E0 ; E with Ž E0 . ) 0 such that P Ž s . E s E _ E0
for some s ) 0.
Ž 1.12.
Then for all f g L1 we have
HX_E P Ž s . f d q HE P Ž s . f d 0
s
HX_Ž E_E . Ž P Ž s . Ž f 1
X_E
. q P Ž s . Ž f 1 E . . d
0
F
HXP Ž s . Ž f 1
X_E
. d s H
X_E
f d .
Using induction argument we obtain n
HX_E P Ž ns . f d q Ý HE P Ž ks . f d F HX_E f d ks1
0
for f g L1. Since the operator P Ž s . is conservative, the sum on the left hand side is infinite and consequently HX _ E f d s ⬁, which is impossible. Thus condition Ž1.12. does not hold and P Ž t . E s E for all t ) 0. Remark 2. If we assume that Ž X, ⌺, . is a finite measure space and P Ž t .4t G 0 is a partially integral Markov semigroup such that P Ž t .1 X s 1 X for all t G 0, then from condition Ž1.11. follows easily that P Ž t .1 E s 1 E for all t ) 0, where 1 E denotes the characteristic function of the set E. In this case we have the following alternative: the operator P Ž t . is a pre-Harris operator for every t G 0 or there exists a set E g ⌺ such that Ž E . ) 0, Ž X _ E . ) 0, and P Ž t .1 E s 1 E for all t ) 0. Proof of Theorem 1. Let us observe that supp f# s X. This follows immediately from the fact that P Žsupp f#. s supp f#. Define a new measure space Ž X, ⌺, . with d s f# d and consider the operators P Ž t . f s Ž 1rf# . P Ž t . Ž f ⭈ f# . ,
t ) 0.
Ž 1.13.
Then P Ž t .4t G 0 is also a partially integral Markov semigroup on the space L1 Ž X, ⌺, .. It is easy to observe that the semigroup P Ž t .4t G 0 is asymptotically stable if and only if the semigroup P Ž t .4t G 0 is asymptotically stable. Thus it is sufficient to show that the semigroup P Ž t .4t G 0 is asymptotically
PICHOR ´ AND RUDNICKI
674
stable. From definition of the measure it follows that Ž X . s 1. From Ž1.13. we have P Ž t .1 X s 1 X , t ) 0 and consequently the operator P Ž t ., t ) 0 is conservative and P Ž t . X s X, t ) 0. According to Lemma 1 the operator P Ž t . is a Harris operator for every t G 0. Fix t 0 ) 0 and denote P s P Ž t 0 .. It is easy to check that the semigroup P Ž t .4t G 0 is asymptotically stable if and only if P n f ª Hf d for f g L1 Ž X, ⌺, .. In order to check asymptotic stability of the operator P we use some properties of Harris operators w6x. Denote ⌺ d s A g ⌺ : P n 1 A s 1 B n , for n s 1, 2, . . . 4 .
Ž 1.14.
Then ⌺ d is an atomic -algebra. We claim that ⌺ d s X, ⭋4 . In order to show this fact we introduce an auxiliary function
A Ž t . s Ž P Ž t . A .
for t G 0,
A g ⌺.
Ž 1.15.
Let 0 - s - t. Since the operator P Ž t y s . is a Markov operator and P Ž t .1 A F 1 P Ž t . A , we have
A Ž s . s
HX1
P Ž s. A
d s
HXP Ž t y s . 1
P Ž s. A
d F Ž P Ž t . A . s A Ž t . .
Ž 1.16. This implies that AŽ t . is a non-decreasing function of t. Let A g ⌺ d and P n 1 A s 1 B n. Since P is a Markov operator we have Ž Bn . s Ž A. and AŽ nt 0 . s AŽ0. for n G 0. Consequently AŽ t . s AŽ0. for t G 0 and A g ⌺ d . Since
H1
A
d s P Ž t . 1 A d F 1 P Ž t . A d s 1 A d
H
H
H
we have P Ž t .1 A s 1 P Ž t . A for t G 0 and A g ⌺ d . This implies that P Ž t . A g ⌺ d for t G 0 and A g ⌺ d . Now let E g ⌺ d be an atom and let ␦ ) 0 be such a number that 5 P Ž s. 1E y 1E 5 - Ž E.
Ž 1.17.
for s F ␦ . From P Ž t .1 E s 1 P Ž t . E and Ž1.17. it follows that P Ž s . E l E / ⭋. Since P Ž s . E g ⌺ d and E is an atom, we have P Ž s . E s E for s F ␦ . Consequently P Ž t . E s E for t G 0 and E g ⌺ d . This implies that ⌺ d s X, ⭋4 . Since ⌺ d is a trivial -algebra, from w6, formula Ž8.7.x it follows that P n f ª Hf d for f g L1 Ž X, ⌺, . which completes the proof. Remark 3. The assumption in Theorems 1 and 2 that the semigroup P Ž t .4t G 0 is a partially integral is essential. Let X s S 1 be a unit circle on
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
675
the complex plain with centre z 0 s 0, let ⌺ s BŽ X . be the -algebra of Borel subsets of X, and let be the arc-Lebesgue measure on X. Then the Markov semigroup P Ž t .4t G 0 given by P Ž t . f Ž z . s f Ž ze i t . has only one invariant density, but it is not asymptotically stable.
2. TRANSPORT EQUATIONS Now we give some examples of partial differential equations which generate Markov semigroups. We also give some applications of Theorem 1 to these equations. In this section X s ⺢ d, ⌺ s BŽ X . is the -algebra of Borel subsets of X and is the Lebesgue measure on X. We write d s dx. 2.1. Fokker᎐Planck Equation In the space X the Fokker᎐Planck equation has the form
⭸u ⭸t
d
s
Ý i , js1
⭸ 2 Ž i j Ž x . u . ⭸ xi ⭸ x j
d
y
Ý is1
⭸ Ž ai Ž x . u . ⭸ xi
,
u Ž x, 0 . s ¨ Ž x . . Ž 2.1.
We assume that the functions i j and a i are bounded and sufficiently smooth. We also assume that d
Ý
i j Ž x . i j G ␣ < < 2
i , js1
for some ␣ ) 0 and every g ⺢ d and x g X. The solution of this equation describes the distribution of a diffusion process. This equation generates a Markov semigroup given by P Ž t . ¨ Ž x . s uŽ x, t ., where ¨ Ž x . s uŽ x, 0.. The Markov semigroup generated by the Fokker᎐Planck equation is an integral semigroup. That is, PŽ t. f Ž x. s
HXq Ž t , x, y . f Ž y . dy,
t ) 0.
Since the kernel q is positive we have P Ž t . E s X for t ) 0 and every measurable set E with positive Lebesgue measure. This implies that E s X is the only invariant set with respect to the semigroup P Ž t .4t G 0 . Thus this semigroup is asymptotically stable if and only if it has an invariant density f# Ži.e., Af# s 0, where A is the infinitesimal generator of the semigroup P Ž t .4t G 0 ..
PICHOR ´ AND RUDNICKI
676 2.2. Liou¨ ille Equation
If we assume that i j ' 0 in Eq. Ž2.1., then we obtain the Liou¨ ille equation d ⭸u ⭸ sy Ý Ž 2.2. Ž ai Ž x . u . . ⭸t is1 ⭸ x i As in the previous example, Eq. Ž2.2. generates a Markov semigroup given by P Ž t . ¨ Ž x . s uŽ x, t ., where ¨ Ž x . s uŽ x, 0.. The semigroup P Ž t .4t G 0 can be given explicitly. Namely, for each x g X denote by t x the solution x Ž t . of the equation x⬘ Ž t . s a Ž x Ž t . .
Ž 2.3.
with the initial condition x Ž0. s x. Then P Ž t . f Ž x . s f Ž yt x . det
d dx
yt x
for f g L1 Ž X . .
Ž 2.4.
Equation Ž2.2. has the following interpretation. In the space X we consider the movement of points given by Eq. Ž2.3.. We look at this movement statistically; that is, we consider the evolution of densities of the distribution of points. Then this evolution is described by Eq. Ž2.2.. The semigroup generated by Eq. Ž2.2. is not partially integral and not asymptotically stable but, as we show in next examples, stochastic perturbations of this semigroup can lead to asymptotically stable semigroups. 2.3. Transport Equations If the equation ⭸⭸ut s Au generates a Markov semigroup S Ž t .4t G 0 , K is a Markov operator, and ) 0, then the equation
⭸u ⭸t
s Au y u q Ku
Ž 2.5.
also generates a Markov semigroup. Let A f s Af y f q Kf. Then Eq. Ž2.5. can be rewritten as the evolution equation u⬘ Ž t . s A u, Ž 2.6. where the solution u is a function from w0, ⬁. to L1 Ž X . and u satisfies the initial condition uŽ0. s f, f g L1 Ž X .. From the Phillips perturbation theorem w5x, Eq. Ž2.6. generates a continuous semigroup of Markov operators P Ž t .4t G 0 on L1 Ž X . given by P Ž t . f s u Ž t . s ey t
⬁
Ý nSn Ž t . f , ns0
Ž 2.7.
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
677
where S0 Ž t . s SŽ t . and Snq 1 Ž t . f s
t
H0 S Ž t y s . KS Ž s . f ds, n
n G 0.
Ž 2.8.
The semigroup P Ž t .4t G 0 satisfies the integral equation P Ž t . f s ey t S Ž t . f q
t y s
H0 e
S Ž s . KP Ž t y s . f ds.
Ž 2.9.
If the semigroup S Ž t .4t G 0 is partially integral or the operator K is partially integral then from Ž2.9. it follows that the semigroup P Ž t .4t G 0 is partially integral. From Ž2.9. and continuity of the semigroups SŽ t .4t G 0 and P Ž t .4t G 0 it follows that for a measurable set E we have P Ž t . E ; E for all t G 0 if and only if KE ; E and SŽ t . E ; E for all t G 0. In particular, if A is the operator from Eq. Ž2.1. then P Ž t .4t G 0 is asymptotically stable if and only if it has an invariant density f# Ži.e., A f# s 0.. In many applications A is the operator from Eq. Ž2.2. and the Markov operator K corresponds to some transition probability function P Ž x, E .; i.e., K *1 E Ž x . s P Ž x, E . for x g X and E g ⌺, where K * denotes the adjoint operator of K. We recall that P : X = ⌺ ª w0, 1x is a transition probability function if for each x g X the function mŽ E . s P Ž x, E . is a probabilistic measure and for each E g ⌺ the function f Ž x . s P Ž x, E . is ⌺-measurable. In this case Eq. Ž2.5. has an interesting probabilistic interpretation. Consider a collection of particles moving under the action of the equation x⬘ s aŽ x .. This motion is modified in the following way. In every time interval w t, t q ⌬ t x a particle with the probability P Ž x, E . ⌬ t q oŽ ⌬ t . changes its position from x to a point from the set E. Then any solution of Ž2.5. is the probability density function of the position of the particle at time t. Note that if the operator K is given by transition probability function P then KE ; E iff P Ž x, E . s 1 for a.e. x g E. In the next section we consider two examples of random movement of this type. In these examples both the semigroup SŽ t .4t G 0 and the operator K are singular Žhave no integral parts. but the semigroup P Ž t .4t G 0 is partially integral. Moreover we give sufficient conditions for asymptotic stability of these semigroups.
3. FURTHER APPLICATIONS In this section X is an open subset of ⺢ d, BŽ X . is the -algebra of Borel subsets of X, and is the Lebesgue measure on X.
PICHOR ´ AND RUDNICKI
678 3.1. Jump Process
Now we consider a jump process connected with an iterated function system. Let Ti : X ª X, for i s 1, . . . , k, be a sequence of continuously differentiable transformations. Let pi : X ª w0, 1x, i s 1, . . . , k, be a sequence of continuous functions such that Ý kis1 pi Ž x . s 1 for each x g X. As before we consider particles which move along the solutions of the equation x⬘ s aŽ x .. At any time interval w t, t q ⌬ t x a particle with the probability pi Ž x . ⌬ t q oŽ ⌬ t . jumps from the point x to Ti Ž x .. Assume that det TiX Ž x . / 0 for almost every x. Then the transformations Ti are nonsingular; that is, if Ž E . s 0 then ŽTiy1 Ž E .. s 0. Let K T i : L1 Ž X . ª L1 Ž X . be the Frobenius᎐Perron operator corresponding to the transformation Ti , i.e., the operator satisfying the condition
HE K
Ti
f Ž x . dx s
HT
y1 i Ž E.
f Ž x . dx
for every measurable set E and f g L1 Ž X, BŽ X ., .. Then Eq. Ž2.5. with the operator K given by the formula k
Kf Ž x . s
Ý K T Ž f Ž x . pi Ž x . .
Ž 3.1.
i
is1
describes the evolution of the densities of the distribution of the above jump process. Assume that t Ž X . ; X for all t G 0. Then Eq. Ž2.2. generates a semigroup SŽ t .4t G 0 of Markov operators on the space L1 Ž X, BŽ X ., ., given by S Ž t . f Ž x . s 1 X Ž yt x . f Ž yt x . det
d dx
yt x ,
Ž 3.2.
where t x denote the solution x Ž t . of the equation x⬘ Ž t . s a Ž x Ž t . . with the initial condition x Ž0. s x. Let P Ž t .4t G 0 be the semigroup generated by Eq. Ž2.5. and let E be a measurable set. We have P Ž t . E ; E for all t G 0 if and only if Ti Ž E . ; E for all i s 1, . . . , k and t Ž E . ; E for all t G 0. PROPOSITION 1. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. Let Ž i1 , . . . , i d . be a gi¨ en sequence of integers from the set 1, . . . , k 4 . Let x 0 g X be a gi¨ en point and let x j s Ti jŽ x jy1 . for j s 1, . . . , d. Set X
X
¨ j s Ti dŽ x dy1 . ⭈⭈⭈ Ti j Ž x jy1 . a Ž x jy1 . y a Ž x d .
679
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
for j s 1, . . . , d. Assume that pi jŽ x jy1 . ) 0 for all j s 1, . . . , d and suppose that the ¨ ectors ¨ 1 , . . . , ¨ d are linearly independent. Then the semigroup P Ž t .4t G 0 is asymptotically stable. Proof. According to Theorem 1 and Remark 1, it is sufficient to prove that the semigroup P Ž t .4t G 0 is partially integral. We check that for some t ) 0, the operator P Ž t . is partially integral. Precisely, we show that there exist t ) 0, ) 0, and neighbourhoods U of x 0 and V of x d such that the operator P Ž t . is partially integral with the kernel k Ž x, y . G 1 V Ž x .1 U Ž y .. Let y g X and t ) 0. Let ⌬ t s s Ž 1 , . . . , d . : i ) 0, 1 q ⭈⭈⭈ qd F t 4 . Define a function y, t on the set ⌬ t by
y , t Ž . s ty 1y 2y ⭈⭈⭈ y d (Ti d ( d ( ⭈⭈⭈ (Ti 2 ( 2 (Ti1 ( 1Ž y . . Let M be the matrix with columns ¨ 1 , . . . , ¨ d . It is easy to check that lim lim
dy , t Ž . d
yªx 0 tª0
s M.
Ž 3.3.
Since det M / 0, we have det
dy , t Ž . d
/0
Ž 3.4.
for y sufficiently close to x 0 and t sufficiently small. Let S d Ž t . be given by Ž2.8.. Then from Ž2.7. it follows that P Ž t . f G ey t dS d Ž t . f for f G 0. Let K Ž t, . be the operator given by K Ž t , . s S Ž t y 1 y 2 y ⭈⭈⭈ yd . KS Ž d . ⭈⭈⭈ KS Ž 1 . and let K *Ž t, . be the adjoint operator of K Ž t, .. Then for every measurable set E we have y t d
HEP Ž t . f Ž x . dx G e
s ey t d
H⌬ HE K Ž t , . f Ž y . dy d t
H⌬ HX f Ž y . K * Ž t , . 1
E
Ž y . dy d .
Ž 3.5.
t
We have K * Ž t , . s S* Ž 1 . K * ⭈⭈⭈ S* Ž d . K *S* Ž t y 1 y 2 y ⭈⭈⭈ yd . , where S*Ž . and K * are the adjoint operators of SŽ . and K. Since S*Ž . f Ž y . s f Ž y ., K *f Ž y . s Ý kis1 pi Ž y . f ŽTi Ž y .., and pi jŽ x jy1 . ) 0, there
PICHOR ´ AND RUDNICKI
680
exists a constant c1 ) 0 such that K * Ž t , . 1 E Ž y . G c1 1 E Ž y , t Ž . .
Ž 3.6.
for y sufficiently close to x 0 and t sufficiently small. From Ž3.5. and Ž3.6. it follows that there exist a constant c 2 and a neighbourhood U of x 0 such that
HEP Ž t . f Ž x . dx G c HU f Ž y . H⌬ 1 Ž E
2
y, t
Ž . . d dy.
Ž 3.7.
t
Substituting x s y, t Ž . to Ž3.7. and using Ž3.4. we obtain
HE P Ž t . f Ž x . dx G HU f Ž y . H
1 E Ž x . dx dy,
y, t Ž ⌬ t .
Ž 3.8.
where is a positive constant. Now decreasing t and the neighbourhood U of x 0 we find a neighbourhood V of x d such that V ; y, t Ž ⌬ t . for y g U. This and Ž3.8. imply that PŽ t. f Ž x. G
HX1
V
Ž x . 1U Ž y . f Ž y . dy,
which completes the proof. Now, consider the case when X is an open subset of ⺢ and we have only one transformation T : X ª X. We assume that T is continuously differentiable and T ⬘Ž x . / 0 for almost every x. Now K is the Frobenius᎐Perron operator corresponding to the transformation T. Equation Ž2.5. takes the form
⭸u ⭸t
q u s y
⭸ Ž au . ⭸x
q Ku.
Ž 3.9.
Assume that t Ž X . ; X for all t G 0. Let SŽ t .4t G 0 and P Ž t .4t G 0 be the semigroups generated by Eqs. Ž2.2. and Ž2.5., respectively. In this case we can state Proposition 1 in the following way. COROLLARY 1. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. If aŽT Ž x .. / T ⬘Ž x . aŽ x . for some x g X, then the semigroup P Ž t .4t G 0 is asymptotically stable. EXAMPLE 1. Consider the equation
⭸u ⭸t
Ž x, t . q
⭸u ⭸x
Ž x, t . q u Ž x, t . s 2 u Ž 2 x, t . ,
x ) 0,
t G 0,
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
681
with the boundary and initial conditions: uŽ0, t . s 0 and uŽ x, 0. s f Ž x .. This equation is of the form Ž3.9. with X s Ž0, ⬁., a ' 1, s 1, Kf Ž x . s 2 f Ž2 x ., and T Ž x . s 2x . We show that the Markov semigroup P Ž t .4t G 0 generated by this equation is asymptotically stable. First we check that X is the only invariant set with respect to the semigroup P Ž t .4t G 0 . Indeed, assume that P Ž t . E ; E for some measurable set E and all t G 0. Then SŽ t . E ; E for all t G 0 and T Ž E . ; E. Since SŽ t . f Ž x . s f Ž x y t .1 w t, ⬁.Ž x ., we have E s Ž ␣ , ⬁. for some ␣ G 0. Now the condition T Ž E . ; E implies that E s X. Then the function f# given by f# Ž x . s
⬁
Ý an ey2
n
x
,
ns0
where n
a0 s 1
and
Ž y2. an s 2 Ž 2 y 1 . Ž 2 y 1 . ⭈⭈⭈ Ž 2 n y 1 .
for n G 1,
is invariant. Since T ⬘Ž x . / 1, according to Corollary 1 the semigroup P Ž t .4t G 0 is asymptotically stable. 3.2. Randomly Controlled Dynamical System Let Yt be a continuous time Markov chain on the phase space ⌫ s 1, . . . , k 4 , k G 2, such that the transition probability from the state j to the state i / j in time interval ⌬ t equals pi j ⌬ t q oŽ ⌬ t .. We assume that pi j ) 0 for all i / j. Let a be a d-dimensional vector function defined on X = ⌫. Let 0 be a d-dimensional random variable independent of Yt . Consider the stochastic differential equation d t s a Ž t , Yt . dt. The pair Ž t , Yt . constitutes a Markov process on X = ⌫. The process t describes the movement of points under the action of k dynamical systems x⬘ s aŽ x, i ., i s 1, . . . , k. The Markov chain Yt decides which dynamical systems acts at time t. We assume that the random variable 0 has an absolutely continuous distribution. Then the random variable t has an absolutely continuous distribution for each t ) 0. Define the function u by the formula Prob Ž Ž t , Yt . g E = i 4 . s
HEu Ž x, i , t . dx.
We assume that a j Ž⭈, i . are continuously differentiable and bounded functions for all i, j. Denote by A i the differential operators d
Ai f s y
Ý js1
⭸ Ž a j Ž x, i . f . ⭸ xj
.
Ž 3.10.
682
PICHOR ´ AND RUDNICKI
Let pii s yÝ j/ i pji and denote by M the matrix w pi j x. We use the notation u i Ž x, t . s uŽ x, i, t . and u s Ž u1 , . . . , u k . is a vertical vector. Then the vector u satisfies the equation
⭸u ⭸t
s Mu q Au,
Ž 3.11.
where Au s Ž A1 u1 , . . . , A k u k . is also a vertical vector. Assume that for each 1 F i F k and for all t G 0 we have ti Ž X . ; X. Then the operator A i generates a semigroup SŽ t .Ž i .4t G 0 of Markov operators on the space L1 Ž X, BŽ X ., .. Let BŽ X = ⌫ . be the -algebra of Borel subsets of X = ⌫ and let m be the product measure on BŽ X = ⌫ . given by mŽ B = i4. s Ž B . for each B g BŽ X . and 1 F i F k. The operator A generates a continuous semigroup SŽ t .4t G 0 of Markov operators on the space L1 Ž X = ⌫, BŽ X = ⌫ ., m. given by the formula S Ž t . f s Ž S Ž t . Ž 1. f 1 , . . . , S Ž t . Ž k . f k . , where f i Ž x . s f Ž x, i . for x g X, 1 F i F k. Now, let be a constant such that s max yp11 , . . . , ypk k 4 and K s y1 M q I. Then Eq. Ž3.11. can be written in the form
⭸u ⭸t
s Au y u q Ku
Ž 3.12.
and the matrix K is a Markov operator on L1 Ž X = ⌫, BŽ X = ⌫ ., m.. Let SŽ t .4t G 0 and P Ž t .4t G 0 be the semigroups generated by the equation ⭸u Ž . ⭸ t s Au and Eq. 3.12 , respectively. A measurable set E ; X = ⌫ is invariant with respect to the semigroup P Ž t .4t G 0 if and only if E s E0 = ⌫ and
ti Ž E0 . s E0
for t G 0 and
i s 1, . . . , k.
Ž 3.13.
Let Ž i1 , . . . , i dq1 . be a sequence of integers from the set 1, . . . , k 4 . For x g X and t ) 0 we define the function x, t on the set ⌬ t by i dq 1 id i2 i1 x , t Ž 1 , . . . , d . s ty 1 y 2 y ⭈⭈⭈ y d ( d ( ⭈⭈⭈ ( 2 ( 1 Ž x . .
PROPOSITION 2. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. Suppose that for some x 0 g X, t 0 ) 0, and 0 g ⌬ t 0 we ha¨ e det
d x 0 , t 0Ž 0 . d
/ 0.
Then the semigroup P Ž t .4t G 0 is asymptotically stable.
Ž 3.14.
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
683
Proof. The proof is similar to the proof of Proposition 1 and we only sketch it. Since the elements of the matrix K are positive constants, there exists a constant c such that
HEP Ž t . f Ž x, i
dq 1
. dx G cH f Ž y, i1 . H 1 E Ž y , t Ž . . d dy ⌬t
X
Ž 3.15.
for every measurable set E and all t from some neighbourhood of t 0 . From Ž3.14. and Ž3.15. we conclude that there exist ) 0 and neighbourhoods U of x 0 and V of x 0 , t 0Ž 0 . such that P Ž t . f Ž x, i dq 1 . G
HX1
V
Ž x . 1U Ž y . f Ž y, i1 . dy,
which proves that the semigroup P Ž t .4t G 0 is partially integral. Consider the case k s d q 1. Set ¨ j Ž x . s aŽ x, j . y aŽ x, d q 1. for j s 1, . . . , d and x g X. If we assume that for some point x 0 g X the vectors ¨ 1Ž x 0 ., . . . , ¨ d Ž x 0 . are linearly independent then condition Ž3.14. holds for t 0 sufficiently small. Indeed, denote by W the matrix with columns ¨ 1Ž x 0 ., . . . , ¨ d Ž x 0 .. Then det W / 0. Moreover, one can check that lim lim
xªx 0 tª0
dx , t Ž . d
s W,
which implies that there is a sufficiently small t 0 such that condition Ž3.14. holds. Now, assume that X is an open subset of ⺢ and ⌫ s 1, 24 . Consider the differential operators A j f Ž x . s y dxd Ž aŽ x, j . f Ž x .., j s 1, 2 and assume that tj Ž X . ; X for j s 1, 2 and for all t ) 0. Moreover assume that p12 ) 0 and p 21 ) 0. Suppose that for some point x 0 g X we have aŽ x 0 , 1. / aŽ x 0 , 2.. Then there is a sufficiently small t 0 such that condition Ž3.14. holds. If P Ž t .4t G 0 is the semigroup generated by Eq. Ž3.12. then according to Proposition 2 we have the following corollary. COROLLARY 2. Assume that the semigroup P Ž t .4t G 0 has a non-zero in¨ ariant function and has no non-tri¨ ial in¨ ariant sets. Moreo¨ er suppose that for some point x 0 g X we ha¨ e aŽ x 0 , 1. / aŽ x 0 , 2.. Then the semigroup P Ž t .4t G 0 is asymptotically stable. EXAMPLE 2. Consider the following system of equations:
⭸ u1 ⭸t ⭸ u2 ⭸t
s ypu1 q pu 2 y s pu1 y pu 2 y
⭸ Ž a Ž x, 1 . u1 . ⭸x
⭸ Ž a Ž x, 2 . u 2 . ⭸x
,
Ž 3.16. .
PICHOR ´ AND RUDNICKI
684
Here p ) 0 and the functions aŽ x, 1. and aŽ x, 2. are continuously differentiable and bounded. We assume that there are some constants c ) 0, L ) 0 such that aŽ x, 1. G c, aŽ x, 2. F yc for all x g ⺢ and
Ž sign x .
1 a Ž x, 1 .
q
1 a Ž x, 2 .
for < x < G L,
Gc
Ž 3.17.
where sign x s xr< x <. Then the semigroup P Ž t .4t G 0 generated by Ž3.16. is asymptotically stable. Indeed, from Ž3.13. it follows immediately that the set ⺢ = 1, 24 is the only invariant set with respect to the semigroup P Ž t .4t G 0 . Let
½
b Ž x . s exp yp
H0
x
1 a Ž s, 1 .
q
1 a Ž s, 2 .
5
ds .
From Ž3.17. it follows that the function f#Ž x, i . s < a bŽ Žx,x i.. < is integrable. Since f# is also a stationary solution of Ž3.16. and aŽ x, 1. ) 0 ) aŽ x, 2., the semigroup P Ž t .4t G 0 is asymptotically stable. REFERENCES 1. K. Baron and A. Lasota, Asymptotic properties of Markov operators defined by Volterra type integrals, Ann. Polon. Math. 58 Ž1993., 161᎐175. 2. W. Bartoszek and T. Brown, On Frobenius᎐Perron operators which overlap supports, Bull. Polish Acad. Math. 45 Ž1997., 17᎐24. 3. S. Chandrasekhar and G. Munch, The theory of fluctuations in brightness of the ¨ Milky-Way, Astrophys. J. 125, 94᎐123. 4. O. Diekmann, H. J. A. Heijmans, and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 Ž1984., 227᎐248. 5. N. Dunford and J. T. Schwartz, ‘‘Linear Operators, Part I,’’ Interscience, New York, 1968. 6. S. R. Foguel, ‘‘The Ergodic Theory of Markov Processes,’’ Van Nostrand Reinhold, New York, 1969. 7. A. Lasota and M. C. Mackey, ‘‘Chaos, Fractals and Noise. Stochastic Aspects of Dynamics,’’ Springer Applied Mathematical Sciences, Vol. 97, Springer-Verlag, New York, 1994. 8. J. Łuczka and R. Rudnicki, Randomly flashing diffusion: asymptotic properties, J. Statist. Phys. 83 Ž1996., 1149᎐1164. 9. M. C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 Ž1994., 89᎐109. 10. J. Malczak, An application of Markov operators in differential and integral equations, Rend. Sem. Mat. Uni¨ . Pado¨ a 87 Ž1992., 281᎐297. 11. K. Pichor, ´ Asymptotic stability of a partial differential equation with an integral perturbation, Ann. Polon. Math. 68 Ž1998., 83᎐96. 12. K. Pichor ´ and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl. 215 Ž1997., 56᎐74.
MARKOV SEMIGROUPS AND TRANSPORT EQUATIONS
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13. K. Pichor ´ and R. Rudnicki, Asymptotic behaviour of Markov semigroups and applications to transport equations, Bull. Polish Acad. Math. 45 Ž1997., 379᎐397. 14. R. Rudnicki, Asymptotic behaviour of an integro-parabolic equation, Bull. Polish Acad. Math. 40 Ž1992., 111᎐128. 15. R. Rudnicki, Asymptotic behaviour of a transport equation, Ann. Polon. Math. 57 Ž1992., 45᎐55. 16. R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Math. 43 Ž1995., 245᎐262. 17. J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Polish Acad. Math. 44 Ž1996., 230᎐252.