Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications

ASYMPTOTIC STABILITY OF EVOLUTION SYSTEMS OF PROBABILITY MEASURES FOR NONAUTONOMOUS STOCHASTIC SYSTEMS: THEORETICAL RESULTS AND APPLICATIONS RENHAI WANG, TOM´ AS CARABALLO, AND NGUYEN HUY TUAN∗ Abstract. The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to Da Prato and R¨ockner [11] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future. 1. Introduction 1.1. Statement of problems. A basic approach to look at the asymptotic stability of stochastic systems with noise perturbations is to consider the limiting stability of their invariant probability measures with respect to the noise intensity. Recently, this kind of the limiting stability of invariant probability measures of time homogeneous transition semigroups was discussed in the literature for autonomous stochastic lattice systems/ODEs/PDEs, see e.g., [5, 6, 7, 14, 15]. As far as the authors can find, currently, there are no results reported in the literature on the limiting stability of an evolution system of probability measures (an extension of invariant measures from autonomous to nonautonomous developed by Da Prato and R¨ockner [10, 11]) of time inhomogeneous transition operators for nonautonomous stochastic systems. 1.2. General framework and theoretical results. The goal of the present work is to initiate a program of studying limiting stability of evolution systems of probability measures of time inhomogeneous transition operators. We will establish a general setting on the limiting stability of evolution system of probability measures for an abstract time inhomogeneous transition operator on a Polish space. More specifically, we will show, under certain conditions, that the limit of every evolution system of probability measures (if exists) must be an evolution system of probability measures of the limiting transition operator. 2010 Mathematics Subject Classification. 37L30,37L40,37L55, 35B40,35B41,60H10. Key words and phrases. Invariant measure; tightness; Limit measure; nonlinear noise; lattice system. ∗Corresponding author: Nguyen Huy Tuan (E-mail: nguy[email protected]). 1 2 RENHAI WANG, TOM´ AS CARABALLO, AND NGUYEN HUY TUAN∗ Let (X,k·kX) be a Polish space, P(X) be the set of all probability measures on X,B(X) be the Borel σ-algebra on X, and Bb(X) (Cb(X)) the space of all bounded Borel (continuous) functions on X. Assume {X(t, τ, x), t ⩾τ∈R}on Xis a unique noise driven stochastic process with initial value x∈Xand the noise intensity ∈(0,˙) for any ˙ > 0. For ϕ∈Bb(X), Λ ∈B(X) and η∈ P(X), we define a transition operator (P τ,t)t⩾τby (P τ,tϕ)(x) = E[ϕ(X(t, τ, x))], a transition probability function P τ,t(x, Λ) = P{ω∈Ω : X(t, τ, x)∈Λ}and the adjoint operator (Q τ,t)t⩾τof (P τ,t)t⩾τby Q τ,tη(Λ) = RXP τ,t(x, Λ)η(dx). As in Da Prato and R¨ockner [10, 11], we say {η t}t∈R⊆ P(X) is an evolution system of probability measures of (P τ,t)t⩾τif Q τ,tητ=ηtfor all t⩾τ∈R. For some technical reasons, the following assumption is needed when we discuss the limiting stability of evolution systems of probability measures of time inhomogeneous transition operators. CIP(Convergence in Probability). For each compact set K⊆X,τ∈R,t⩾τ,0∈(0,˙] and δ > 0, lim →0 sup x∈K Pω∈Ω : kX(t, τ, x)−X0(t, τ, x)kX⩾δ= 0.(1.1) Theorem 1.1. (Theoretical result I) Assume CIP holds and (P0 τ,t)t⩾τis Feller. Let {η0 t}t∈R be a family of probability measures on Xfor 0∈[0,˙], and {ηn t}t∈Rbe an evolution system of probability measures of (Pn τ,t)t⩾τon Xwith n→0. If ηn t→η0 tweakly for each t∈R, then {η0 t}t∈Rmust be an evolution system of probability measures of (P0 τ,t)t⩾τ. Remark 1.2. (i) We only require (P τ,t)t⩾τis Feller at =0. (ii) The convergence in probability in (1.1) is not necessary to be uniform for t. But for most stochastic systems, (1.1) can be proved uniformly for t:lim→0supx∈KPω∈Ω : supt∈[τ,τ+T]kX(t, τ, x)−X0(t, τ, x)kX⩾δ= 0. In applications, nin Theorem 1.1 may depend on t, we also provide the following results. Theorem 1.3. (Theoretical result II) Assume CIP holds and (P0 τ,t)t⩾τis Feller. Let {η0 t}t∈R be a family of probability measures on Xfor 0∈[0,˙], and {ηn t}t∈Rbe an evolution system of probability measures of (Pn τ,t)t⩾τon Xwith n→0. For each t∈R, if there exists a subsequence {nk(t)}∞ k=1 of {n}∞ n=1 such that ηnk(t) t→η0 tweakly, then {η0 t}t∈Rmust be an evolution system of probability measures of (P0 τ,t)t⩾τ. Theorems 1.1 and 1.3 can be viewed as extension versions of limiting stability of invariant measures of time homogeneous transition operators from the autonomous framework to the nonautonomous setting. 1.3. Application of theoretical results. Our abstract results in Theorems 1.1 and 1.3 are expected to be applied to various stochastic ODEs/PDEs/lattice systems with noise perturbations. In particular, we apply these abstract results to a stochastic lattice reaction-diffusion equation driven by a nonlinear noise on Zfor t>τ with τ∈R: du i(t) + λu i(t)dt −νu i(t)−2u i−1(t) + u i+1(t)dt +|u i(t)|p−2|u i(t)|dt =σ(t, u i(t))dW(t),(1.2) ASYMPTOTIC STABILITY OF EVOLUTION SYSTEMS OF PROBABILITY MEASURES 3 with initial condition u i(τ) = u0,i, where λ, ν > 0, p > 2, ⩾0, Wis a two-sided, real-valued Wiener process on (Ω,F,{Ft}t∈R,P) the complete filtered probability space, and the nonlinear function σ(t, ·) : R→Ris locally Lipschitz such that |σ(t, s)|⩽δ|s|+g(t) for a constant δ > 0 and a time-dependent deterministic function gsatisfying certain conditions. We say the noise in (1.2) is a time-dependent nonlinear noise just because the diffusion function σdepends on time t, and is nonlinear in the unknown function u i. Note that lattice systems can be regarded as space discretization versions of PDEs that have many applications in the real world such as electric circuits, pattern formation, propagation of nerve pulse, chemical reaction and others. The existence and limiting stability of invariant probability measures for autonomous stochastic lattice systems has been considered recently in [5, 6, 14, 15, 16]. The reader is referred to [1, 2, 3, 4, 12, 13, 18, 19] for other mathematical topics such as random attractors for stochastic lattice systems. In this paper we study existence and limiting stability of evolution systems of probability measures for the nonautonomous stochastic lattice system (1.2). To our knowledge, it seems that this is the first time to study evolution systems of probability measures for stochastic lattice systems. Theorem 1.4. (Existence of evolution systems of probability measures) If R0 −∞ eλrkg(r)k2dr < ∞, then the transition operator (P τ,t)t⩾τfor (1.2) has an evolution system of probability measures {µ t}t∈Ron `2for any ∈0,√λ 2δ. Let E t=µ t:{µ t}t∈Ris an evolution system of probability measures of (P τ,t)t⩾τfor (1.2)for t∈Rand ∈0,√λ 2δ. Then we discuss the tightness of S∈0,√λ 2δE tand limiting stability of any sequences of E t. Theorem 1.5. (Application of theoretical results) If Rt −∞ eλrkg(r)k2dr < ∞for each t∈R, then we have the following two results. (i) The union S∈0,√λ 2δE tis tight on `2for each t∈R. (ii)For each t∈R, if µn t∈ En twith n→0∈0,√λ 2δ, then there exists µ0 t∈ E0 tand a subsequence {nk(t)}∞ k=1 of {n}∞ n=1 such that µn t→µ0 tweakly for each t∈R. 2. Proof of Theorems 1.1 and 1.3 Proof. We only prove Theorem 1.3. Given τ∈R,s⩾τand ϕ∈Cb(X), it is sufficient to show ZX ϕ(x)(Q0 τ,sη0 τ)(dx) = ZX (P0 τ,sϕ)(x)η0 τ(dx) = ZX ϕ(x)η0 s(dx).(2.1) The first equality is obvious. By the condition for t=s, there exists a subsequence {nk(s)}∞ k=1 of {n}∞ n=1 such that ηnk(s) s→η0 sweakly. Then, for every γ > 0, there exists N1∈Nsuch that ZX ϕ(x)ηnk(s) s(dx)−ZX ϕ(x)η0 s(dx)⩽γfor all k⩾N1.(2.2) Since nk(s)→0and nηnk(s) tot∈Ris an evolution system of probability measures of Pnk(s) τ,t t⩾τ, by the condition again for t=τ, there exists a subsequence {nkj(s, τ)}∞ j=1 of {nk(s)}∞ k=1 such that 4 RENHAI WANG, TOM´ AS CARABALLO, AND NGUYEN HUY TUAN∗ ηnkj(s,τ) τ→η0 τweakly. This implies that for every γ > 0, there exists N2∈Nsuch that ZX (P0 τ,sϕ)(x)ηnkj(s,τ) τ(dx)−ZX (P0 τ,sϕ)(x)η0 τ(dx)⩽γfor all j⩾N2.(2.3) where we have used the Feller property of P0 τ,s. By the assumption CIP we know that for every γ > 0 and δ > 0, there exists N3∈Nsuch that sup x∈K Pnω∈Ω : kXnkj(s,τ)(s, τ, x)−X0(s, τ, x)kX⩾δo⩽γfor all j⩾N3.(2.4) Since ηnkj(s,τ) τ→η0 τweakly, by Prohorov theorem (see [8, 9]) we know nηnkj(s,τ) τo∞ j=1 is tight on X. This means that for every γ > 0, there exists a compact set K(γ, s, τ)⊆Xsuch that ηnkj(s,τ) τ(X/K(γ, s, τ)) < γ for all j∈N.(2.5) By ϕ∈Cb(X) and the compactness of K(γ, s, τ) we know that for every γ > 0, there exists η > 0 such that |ϕ(y)−ϕ(z)|< γ for all y, z ∈K(γ, s, τ) with ky−zkX⩽η. (2.6) Indeed, if (2.6) is false, then there exist γ0>0, yn, zn∈K(γ, s, τ) with kyn−znkX⩽1/n such that |ϕ(yn)−ϕ(zn)|⩾γ0. Since K(γ, s, τ) is a compact subset of X, there exist x∈K(γ, s, τ) and a subsequence {nk}∞ k=1 ⊆ {n}∞ n=1 such that limk→∞ kynk−xkX= 0 . Then one can verify limk→∞ kznk−xkX= 0. Since ϕis continuous, letting k→ ∞ in γ0⩽|ϕ(ynk)−ϕ(znk)|⩽ |ϕ(ynk)−ϕ(x)|+|ϕ(znk)−ϕ(x)|, we find a contradiction γ0⩽0. ASYMPTOTIC STABILITY OF EVOLUTION SYSTEMS OF PROBABILITY MEASURES 5 Since nηnkj(s,τ) tot∈Ris an evolution system of probability measures of Pnkj(s,τ) τ,t t⩾τon X, by (2.4)-(2.6) we find that for all j⩾N:= max{N1, N2, N3}, ZX (P0 τ,sϕ)(x)ηnkj(s,τ) τ(dx)−ZX ϕ(x)ηnkj(s,τ) s(dx) =ZX Eϕ(X0(s, τ, x))ηnkj(s,τ) τ(dx)−ZX Eϕ(Xnkj(s,τ)(s, τ, x))ηnkj(s,τ) τ(dx) ⩽ZX E|ϕ(Xnkj(s,τ)(s, τ, x)) −ϕ(X0(s, τ, x))|ηnkj(s,τ) τ(dx) ⩽ZK(γ,s,τ) E|ϕ(Xnkj(s,τ)(s, τ, x)) −ϕ(X0(s, τ, x))|ηnkj(τ) τ(dx) + 2 sup x∈X|ϕ(x)|ηnkj(s,τ) τ(X/K(γ, s, τ)) ⩽ZK(γ,s,τ) E|ϕ(Xnkj(s,τ)(s, τ, x)) −ϕ(X0(s, τ, x))|ηnkj(s,τ) τ(dx)+2γsup x∈X|ϕ(x)| ⩽ZK(γ,s,τ)Zω∈Ω:kXnkj(s,τ)(s,τ,x)−X0(s,τ,x)kX⩾ηϕ(Xnkj(s,τ)(s, τ, x)) −ϕ(X0(s, τ, x))P(dω)ηnkj(s,τ) τ(dx) +ZK(γ,s,τ)Zω∈Ω:kXnkj(s,τ)(s,τ,x)−X0(s,τ,x)kX<ηϕ(Xnkj(s,τ)(s, τ, x)) −ϕ(X0(s, τ, x))P(dω)ηnkj(s,τ) τ(dx) + 2γsup x∈X|ϕ(x)| ⩽2 sup x∈X|ϕ(x)|ηnkj(s,τ) τ(K(γ, s, τ))Pnω∈Ω : kXnkj(s,τ)(s, τ, x)−X0(s, τ, x)kX⩾ηo +γηnkj(s,τ) τ(K(γ, s, τ))Pnω∈Ω : kXnkj(s,τ)(s, τ, x)−X0(s, τ, x)kX< ηo+ 2γsup x∈X|ϕ(x)| ⩽γ(1 + 4 sup x∈X|ϕ(x)|).(2.7) In order to prove the second equality in (2.1), we consider the following equality: ZX (P0 τ,sϕ)(x)η0 τ(dx)−ZX ϕ(x)η0 s(dx) = ZX (P0 τ,sϕ)(x)η0 τ(dx)−ZX (P0 τ,sϕ)(x)ηnkj(s,τ) τ(dx) +ZX (P0 τ,sϕ)(x)ηnkj(s,τ) τ(dx)−ZX ϕ(x)ηnkj(s,τ) s(dx) +ZX ϕ(x)ηnkj(s,τ) s(dx)−ZX ϕ(x)η0 s(dx).(2.8) Thus, as a result of (2.2)-(2.3) and (2.7)-(2.8) we have ZX (P0 τ,sϕ)(x)η0 τ(dx)−ZX ϕ(x)η0 s(dx)⩽γ(3 + 4 sup x∈X|ϕ(x)|). Since γ > 0 is arbitrary, we thus complete the proof.  3. Proof of Theorem 1.4 Let us consider the Banach space `q=u= (ui)i∈Z:Pi∈Z|ui|q<+∞endowed with the norm kukq=(Pi∈Z|ui|q1/q,if q∈[1,∞), supi∈Z|ui|,if q=∞. 6 RENHAI WANG, TOM´ AS CARABALLO, AND NGUYEN HUY TUAN∗ In particular, we write k · k =k · k2. It is known from Wang [16] that for each τ∈Rand u0∈L2(Ω,Fτ;`2), equation (1.2) has a unique solution which is continuous `2-valued Ft-adapted Markov process u(·, τ, u0)∈L2(Ω, C([τ, ∞), `2)) ∩Lp(Ω, Lp(τ, ∞); `p)). Then we can prove that the transition operator (P τ,t)t⩾τfor u(t, τ, u0) with u0∈`2is Feller, and the process laws hold: P τ,t =P τ,rP r,t and Q τ,t =Q r,tQ τ,r,−∞ < τ ⩽r⩽t < +∞. Next, we derive different uniform estimates of time average of the solutions. Lemma 3.1. If Rt −∞ eλrkg(r)k2dr < ∞for each t∈R, then we have the following results. (i) For any t⩾τ∈Rand 0,√λ 2δ, Eku(t, τ, u0)k2+λ 2Zt τ eλ(r−t)Eku(r, τ, u0)k2dr ⩽eλ(τ−t)Eku0k2+cZt −∞ eλ(s−t)kg(s)k2ds, (3.1) where c > 0is a constant independent of ,tand u0. (ii) For any t∈Rand N3k > −tand ∈0,√λ 2δ, 1 k+tZt −k E[ku(t, τ, u0)k2]dτ +1 k+tZt −kZt τ eλ(r−t)Eku(r, τ, u0)k2drdτ ⩽c (k+t)Eku0k2+cZt −∞ eλ(r−t)kg(r)k2dr, (3.2) where c > 0is a constant independent of ,t,kand u0. (iii) For each t∈Rand bounded set Bof `2, the solutions satisfy lim n→∞ lim τ→−∞ sup ∈0,√λ 2δ sup u0∈BX |i|⩾n E[|u i(t, τ, u0)|2]=0. (iv) For each t∈Rand compact set Kof `2, the solutions satisfy lim n→∞ sup ∈0,√λ 2δ sup N3k>−t sup u0∈K 1 k+tZt −kX |i|⩾n E[|u i(t, τ, u0)|2]dτ = 0. Proof. (i)-(ii) Applying Itˆo’s formula to (1.2), we infer that for any ∈0,√λ 2δ, d dtEku(t)k2+3 2λEku(t)k2⩽ckg(t)k2.(3.3) Multiplying (3.3) by eλt and integrating over (τ, t), we deduce (3.1). Integrating (3.1) with respect to τover (−k, t), we obtain (3.2). (iii)-(iv) By a cut-off technique as used by Wang [16, Lemma 4.2] (see also [17]) one can derive, for any ∈0,√λ 2δ, EhX |i|⩾2n|u i(t, τ, u0)|2i⩽eλ(τ−t)X |i|⩾n|u0,i|2+cZt τ eλ(r−t)X |i|⩾n|gi(r)|2dr +c nZt τ eλ(r−t)Eku(r, τ, u0)k2dr. (3.4) This along with Rt −∞ eλrkg(r)k2dr < ∞and (3.1) completes the proof of (iii). ASYMPTOTIC STABILITY OF EVOLUTION SYSTEMS OF PROBABILITY MEASURES 7 Integrating (3.4) with respect to τover (−k, t), we have 1 k+tZt −k EhX |i|⩾2n|u i(t, τ, u0)|2idτ ⩽1 λ(k+t)X |i|⩾n|u0,i|2+cZt −∞ eλ(r−t)X |i|⩾n|gi(r)|2dr +c n(k+t)Zt −kZt τ eλ(r−t)Eku(r, τ, u0)k2drdτ. This along with Rt −∞ eλrkg(r)k2dr < ∞and (3.2) completes the proof of (iv).  Proof. Proof of Theorem 1.4. The proof is completed by an extended Krylov-Bogolyubov method proposed by Da Prato and R¨ockner [11]. For n∈N, we let χ[−n,n]be the characteristic function of [−n, n]. Then u(t, τ, u0) = ˜u n(t, τ, u0) + ˆu n(t, τ, u0) with ˜u n(t, τ, u0) = χ[−n,n](|i|)u i(t, τ, u0)i∈Z and ˆu n(t, τ, u0) = (1−χ[−n,n](|i|))u i(t, τ, u0)i∈Z. Given δ > 0, l∈Nand m, k ∈Nwith m⩽k, by (ii) and (iv) of Lemma 3.1 there exist nδ l(m)∈Nand C(m)>0 independent on and ksuch that sup ∈0,√λ 2δ 1 k−mZ−m −k Eku(−m, τ, 0)k2dτ ⩽C(m), and sup ∈0,√λ 2δ 1 k−mZ−m −k Ekˆu nδ l(m)(−m, τ, 0)k2dτ < δ 24l. Define Yδ l(m) = u∈`2:ui= 0 for |i|> nδ l(m) and kuk⩽2lpC(m) √δ and Zδ l(m) = nu∈`2:ku−vk⩽1 2lfor some v∈ Yδ l(m)o. Define a probability measure η k,m =1 k−mR−m −kPnω∈Ω : u(−m, τ, 0) ∈ ·odτ on `2. Then by Chebychev’s inequality one can verify η k,m(`2\Zδ l(m)) ⩽δ 22lC(m) 1 k−mZ−m −k Eku(−m, τ, 0)k2dτ +22l k−mZ−m −k Ekˆu nδ l(m)(−m, τ, 0)k2dτ ⩽δ 22l−1.(3.5) Let Zδ(m) = T∞ l=1 Zδ l(m). Since one can verify that Zδ(m) is closed and totally bounded in `2, it is compact in `2. By (3.5) we know η k,m(`2\ Zδ(m)) ⩽P∞ l=1 η k,m(`2\ Zδ l(m)) < δ. Then {η k,m}N3k⩾mis tight on `2for each fixed m∈N. Then there exists η m∈ P(`2) and a subsequence (not relabeled) such that η k,m →η mweakly as k→ ∞. For each fixed t∈R, we choose m∈N such that −m⩽t, and define µ t:= Q −m,tη m. By [11] one can verify that this definition is independent of the choice of m. Then for every fixed t∈Rand for any N3m⩾−τ⩾−t, we have Q τ,tµ τ= (P −m,τ P τ,t)∗η m=Q −m,tη m=µ t. This complete the proof.  8 RENHAI WANG, TOM´ AS CARABALLO, AND NGUYEN HUY TUAN∗ 4. Proof of Theorem 1.5 Proof. Proof of Theorem 1.5. (i) For every δ > 0, t∈R,∈[0,√λ 2δand µ t∈ E t, we shall find a compact set Zδ(t)⊆`2independent of such that µ t(Zδ(t)) >1−δ. Given t∈Rand l∈N, by (i) and (iii) of Lemma 3.1 there exists nδ l(t)∈N,Tδ l(t)⩽tand C(t)>0 independent of and u0 such that sup ∈0,√λ 2δ sup τ⩽t sup u0∈`2 Eku(t, τ, u0)k2dτ ⩽C(t),(4.1) sup ∈0,√λ 2δ sup τ⩽Tδ l(t) sup u0∈`2 Ekˆu nδ l(t)(t, τ, u0)k2<δ 24l.(4.2) Define Yδ l(t) := u∈`2:ui= 0 for |i|> nδ l(t) and kuk⩽2l√C(t) √δ,Zδ l(t) := nu∈`2: ku−vk⩽1 2lfor some v∈ Yδ l(t)oand Zδ(t) := T∞ l=1 Zδ l(t). Note that Zδ(t) is compact in `2. In what follows we prove µ t(Zδ(t)) >1−δ. Denote by Xδ n(t) := Tn l=1 Zδ l(t) for n∈N. Then µ t(Zδ(t)) = µ tT∞ n=1 Xδ n(t)= lim n→∞µ tXδ n(t), and hence there exists N=N(δ, t)∈Nsuch that 0⩽µ t(Xδ n(t)) −µ t(Zδ(t)) ⩽δ/3 for all n⩾N. Note that by (4.1)-(4.2) we have, for all τ⩽Tδ l(t), Z`2 Pnω∈Ω : u(t, τ, x)/∈ Xδ N(t)oµ τ(dx) ⩽ N X l=1 δ 22lC(t)Z`2 Ek˜u nδ l(t)(t, τ, x)k2µ τ(dx) + N X l=1 22lZ`2 Ekˆu nδ l(t)(t, τ, x)k2µ τ(dx) ⩽ N X l=1 δ 22l−1⩽2δ 3.(4.3) By (4.3) and the definition of {µ t}t∈Rof (P τ,t)t⩾τ, we find that for all τ⩽Tδ l(t), µ t(Xδ N(t)) = Z`2 Pnω∈Ω : u(t, τ, x)∈ Xδ N(t)oµ τ(dx) = 1 −Z`2 Pnω∈Ω : u(t, τ, x)/∈ Xδ N(t)oµ τ(dx) ⩾1−2δ 3.(4.4) This concludes the proof of (i). (ii) For every τ∈R,t⩾τ δ > 0, 0∈0,√λ 2δand compact set Kof `2, since we can prove that all uniform estimates of the solutions are uniform for ∈0,√λ 2δ, by a stoping time argument, we can prove lim →0 sup u0∈K Pω∈Ω : ku(t, τ, u0)−u0(t, τ, u0)k⩾δ= 0.(4.5) ASYMPTOTIC STABILITY OF EVOLUTION SYSTEMS OF PROBABILITY MEASURES 9 The proof of (4.5) is quite similar to the autonomous case as in [5, 6, 14, 15], the details are omitted here. 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