Asymptotic exponential stability for diffusion processes driven by stochastic differential equations in duals of nuclear spaces

ASYMPTOTIC EXPONENTIAL STABILITY FOR DIFFUSION PROCESSES DRIVEN BY STOCHASTIC DIFFERENTIAL EQUATIONS IN DUALS OF NUCLEAR SPACES TOM´ AS CARABALLO Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla. Apartado de Correos 1160 41080-Sevilla, SPAIN KAI LIU* Department of Probability and Statistics The University of Sheffield The Hicks Building, Hounsfield Road Sheffield, S3 7RH, UK ABSTRACT The main objective of this paper is to investigate the asymptotic stability for diffusion processes driven by a class of Itˆo stochastic differential equations in duals of nuclear spaces. A coercivity condition imposed on this sort of equation plays the role of an exponential stability criterion. An example is studied to illustrate our theory. AMS Classification: primary 93E03; secondary 60H10. Keywords: Almost sure exponential stability; L2-exponential stability; Stochastic diffusion processes in duals of nuclear spaces. * Author of Correspondence. 1 1. Introduction In the paper we shall study the exponential stability of stochastic diffusion equations in duals of nuclear spaces. These equations naturally arise in the research of chemical reaction-diffusion equations, neurophysiology and turbulence, especially, in the recent river pollution model researches (see [11], [15] and [16]). Roughly speaking, we shall consider the following stochastic diffusion equation: Xt=X0+Zt 0 A(s, Xs)ds +Zt 0 B(s, Xs)dWs(1.1) where A:R+×Φ0→Φ0,B:R+×Φ0→ L(Φ0,Φ0) are two measurable mappings and Wt is a Φ0-valued Wiener process. Here Φ0is the dual space of a certain countably Hilbertian nuclear space and L(Φ0,Φ0) is the space of all bounded linear operators from Φ0into itself. Diffusion equations of the type (1.1) have been studied by a number of authors, for instance, G. Kallianpur and R.L. Wolpert [10], G. Kallianpur and J. Xiong [11], H. Tuckwell [15] and J.B. Walsh [16] among others. The reader is referred to G. Kallianpur and J. Xiong [11] for further details concerning certain properties of the solutions (1.1) and some related topics. In the paper, we are particularly interested in the criteria of exponential stability in the sense of mean square and pathwise with probability one of the strong solutions to the equations (1.1). It is a long history for the investigation of the exponential stability of stochastic differential equations in finite dimensional spaces and, more recently, of stochastic evolution equations in Hilbert spaces. For infinite dimensional case, we should mention U.G. Haussmann [6] (linear case) and A. Ichikawa [7] (semilinear case) for their fundamental work on this aspect. Nevertheless, for nuclear space-valued stochastic differential equation situations, to the best of our knowledge it seems that nobody ever carried out the study of exponential stability either in the sense of mean square or pathwise with probability one. This is the main task in this paper to fill this gap. It is particularly worth pointing out that our approaches, which are devoted to the consideration of the stochastic differential equations in duals of nuclear spaces (1.1), could even be used to extend the results of [6][7] to cover general non-autonomous Hilbert space-valued stochastic differential systems. Firstly, we shall give sufficient conditions for the exponential stability in mean square of the strong solutions to the equations (1.1). Next, we obtain exponential stability of paths with probability one. Our argument is based on a coercivity condition which plays a key role for the existence and uniqueness of the equations (1.1). As a consequence, we will observe how a suitable coercivity condition may be regarded as an exponential stability criterion. The exposition is as follows. In Section 2, we shall briefly collect some notions and notations which are essential for our stability analysis. Section 3 is devoted to the investigation of exponentially asymptotic stability of strong solutions. Finally, in Section 4 we will illustrate the theorems derived in the last section by studying an example. 2 2. Preliminaries In this section we are going to state some basic notions and notations in a suitable way. In particular, the reader is strongly referred to G. Kallianpur and J. Xiong [11] for a systematic and detailed statement concerning the material in this section. Let Φ be a separable Fr´echet space which is a countably Hilbertian space, that is, its topology is given by an increasing separable k·kn,n≥0, of compatible Hilbertian norms. In particular, throughout this paper we suppose Φ is nuclear, precisely, for each n≥0 there exists m > n such that the canonical injection from Φminto Φnis Hilbert-Schmidt. Here Φnis the completion of Φ with respect to k·kn. Let Φ0be the collection of all continuous linear maps from Φ to R, i.e., the dual space of Φ. We could show that {Φn}n≥0is a sequence of decreasing Hilbertian spaces and Φ = ∩∞ n=0Φn. Identifying Φ0 0with Φ0by Riesz’s representation theorem, we denote Φ0 nby Φ−nwith norms k·k−n,n≥0. Then {Φ−n}n≥0is a sequence of increasing Hilbertian spaces, Φ0is sequentially complete and Φ0=∪∞ n=0Φ−n.In the latter case, we shall denote by {φp j} ⊂ Φ a complete orthonormal system, or simply, CONS of Φpand {φ−p j}the CONS of Φ−pconjugate to {φp j}for p≥0. Let θpbe the isometry from Φ−pto Φpsuch that θpφ−p j=φp j,∀j≥1. A class of important examples of countably Hilbertian spaces can be described appropriately as follows. Let Hbe a real separable Hilbert space and A=−La closed densely defined self-adjoint operator on Hsuch that <−Lφ, φ >≤0 for φ∈Dom(L), the domain of L. Let {Tt}be the semigroups on Hdetermined by A. Further assume that some power of the resolvent of Lis a Hilbert-Schmidt operator, i.e., ∃r1such that (λI +L)−r1is Hilbert-Schmidt.(2.1) This condition enables us to prove that there exist 0 ≤λ1≤λ2≤ ··· and {φj} ⊂ H, a CONS of H, such that Lφj=λjφj,for any j≥1. Define Φ =nφ∈H:k(I+L)rφk2 H<∞,∀r∈Ro =½φ∈H: ∞ X j=1 (1 + λj)2r< φ, φj>2 H<∞,∀r∈R¾, and the inner product <·,·>ron Φ by < φ, ψ >r= ∞ X j=1 (1 + λj)2r< φ, φj>H< ψ, φj>H and kφk2 r=< φ, φ >r. Let Φrbe the k·kr-completion of Φ. We then have Φ = \ r Φr,Φ0=[ r Φr 3 and for r≤s,φ∈Φ, kφkr≤ kφksand furthermore Φs⊂Φrwith Φ0=H. Condition (2.1) implies that the injection from Φqinto Φpis Hilbert-Schmidt for q≥p+r1and therefore Φ is a countably Hilbertian nuclear space, simply, CHNS. As usual, we also call the compatible family (Φ, H, Tt) or (Φ, H, L) a special compatible family. We assume throughout that (Ω,F,{Ft}t≥0, P ) is a complete probability space with a right continuous filtration {Ft}t≥0. A map X: Ω →Φ0is a Φ0-valued random variable if it is F/B(Φ0)-measurable, where B(Φ0) is the Borel field of the topological space Φ0(in the sense of strong topology). A family {Xt;t∈R+}of Φ0-valued random variables is called a Φ0-process. In the rest of this paper, we shall concern with Φ0-valued martingales. In particular, we have the following: Definition 2.1. A Φ0-valued process M={Mt}t≥0is a Φ0-martingale with respect to {Ft}t≥0if for each φ∈Φ, Mt[φ] is a martingale with respect to {Ft}. It is called a Φ0-square-integrable-martingale if, in addition, E³Mt[φ]2´<∞,∀φ∈Φ, t ≥0.(2.2) We let M(Φ0) (resp. M2(Φ0)) denote the collection of all Φ0-martingales (resp. Φ0-square- integrable-martingales). We also let M2,c(Φ0) = nM∈ M2(Φ0) : Mt[φ] has a continuous version for each φ∈Φo. Definition 2.2. A continuous (in the sense of strong topology) Φ0-valued stochastic process W= (Wt)t≥0on (Ω,F, P) is called a centered Φ0-Wiener process with Q(·,·) if W satisfies the following three conditions: a). W0= 0 a.s.; b). Whas independent increments, i.e., the random variables Wt1[φ1],(Wt2−Wt1)[φ2],···,(Wtn−Wtn−1)[φn] (2.3) are independent for any φ1,φ2,···,φn∈Φ, 0 ≤t1≤ ··· ≤ tn,n≥1; c). For each t≥0 and φ∈Φ E³eiWt[φ]´=e−tQ(φ,φ)/2(2.4) where Qis a covariance functional, i.e., a positive definite symmetric continuous bilinear form on Φ ×Φ. Clearly, W∈ M2,c(Φ0), {Wt[φ] : φ∈Φ, t ≥0}is a centered Gaussian system and E³Wt[ψ]Ws[φ]´= (s∧t)Q(ψ, φ), ψ, φ ∈Φ, s, t ≥0.(2.5) 4 Definition 2.3. Let Hbe a separable Hilbert space with norm k·kH. A family {Bt(h) : t≥0, h ∈H}of real-valued random variables is called a cyclindrical Brownian motion (c.B.m) on Hwith covariance Σ if Σ is a continuous self-adjoint positive definite operator on Hsuch that the following conditions hold: i). For each h∈Hsuch that h6= 0, <Σh, h >−1/2 HBt(h) is a one-dimensional standard Wiener process; ii). For each t≥0, α1,α2∈Rand f1,f2∈H Bt(α1f1+α2f2) = α1Bt(f1) + α2Bt(f2)a.s.; iii). For each h∈H,{Bt(h)}is an FB t-martingale, where FB t=σ{Bs(h) : s≤t, h ∈H}. {Bt(h) : t≥0, h ∈H}is called a standard H-c.B.m or simply, H-c.B.m. if it is a H-c.B.m. with covariance Σ = I. For each φ∈Φ, let ıφ := Q(φ, ·). Then ıis an injective linear operator from Φ onto a linear subspace R(ı) of Φ0. In particular, for arbitrary v1,v2∈ R(ı), let HQ:= Q(ı−1v1, ı−1v2). Then <·,·>HQis an inner product on R(ı). Let k·kHQbe the norm on R(ı) determined by the inner product <·,·>HQand let HQbe the completion of R(ı) with respect to k · kHQ. Then HQis a separable Hilbert space and HQ⊂Φ0. It could also be shown that there exists a one-to-one correspondence between a Φ0-valued Wiener process Wwith covariance Qand an HQ-c.B.m. B: Wt= ∞ X j=1 Bt(fj)fj(2.6) where {fj}is a CONS of HQ; Bt(v) = lim n→∞ Wt[ı−1vn],∀v∈HQ(2.7) where {vn} ⊂ R(ı) converges to vin HQ. Consider the following stochastic diffusion equation (see [11] for further details on stochastic integral and related properties) Xt=X0+Zt 0 A(s, Xs)ds +Zt 0 B(s, Xs)dWs(2.8) where A:R+×Φ0→Φ0,B:R+×Φ0→ L(Φ0,Φ0) are two measurable mappings and Wtis a Φ0-valued Wiener process. Here L(Φ0,Φ0) denotes the collection of all continuous linear mappings from Φ0into Φ0. Definition 2.4. Let (Ω,F,{Ft}t≥0, P) be the stochastic basis and Wta Φ0-valued Wiener process with covariance function Q. Suppose that X0is a Φ−p-valued random 5 variable such that EkX0k2 −p<∞. Then by a Φ−p-valued strong solution on Ω to the SDE (2.8) for t∈[0, T] we mean a process Xtdefined on Ω such that (a). Xtis a Φ−p-valued Ft-measurable random variable; (b). Xt∈C([0, T],Φ−p), a.s.; (c). There exists a sequence (σn) of bounded stopping times on Ω increasing to infinity such that ∀n≥1 EZT∧σn 0kA(s, Xs)k−qds < ∞,(2.9) and EZT∧σn 0kB(s, Xs)k2 L(2)(HQ,Φ−p)ds < ∞.(2.10) Here L(2)(HQ,Φ−p) denotes the class of all Hilbert-Schmidt operators from HQinto Φ−p and qwill be introduced in the following assumption (H1); (d). The SDE (2.8) is satisfied for all t∈[0, T] and almost all ω∈Ω. If Tis replaced by ∞, we call Xta global strong solution of (2.8). As we are mainly interested in the stability analysis, one always assumes that the equation (2.8) has a unique global strong solution. In particular, for this purpose we shall make the following assumption (H1) [11]: There exists an index p0>0 such that, ∀p≥p0,∃q≥pand a constant K=K(p, q)> 0 such that (D1). (Continuity) ∀t∈R+, the maps v∈Φ−p→A(t, v)∈Φ−qand v∈Φ−p→ B(t, v)∈L(2)(HQ,Φ−p) are continuous; (D2). (Coercivity) ∀t∈R+and v∈Φ−p, we have 2A(t, v)[θpv] + kB(t, v)k2 L(2)(HQ,Φ−p)≤K(1 + kvk2 −p); (2.11) (D3). (Growth) ∀t∈R+and v∈Φ−p, we have kA(t, v)k2 −q≤K(1 + kvk2 −p); (2.12) (D4). (Lipschitz) ∀t∈R+,v1,v2∈Φ−p, we have kA(t, v1)−A(t, v2)k−q≤Kkv1−v2k−p(2.13) and kB(t, v1)−B(t, v2)kL(2)(HQ,Φ−p)≤Kkv1−v2k−p.(2.14) 6 3. The Main Results In this section, we shall devote ourselves to the investigation of exponential stability of the equation (2.8). For simplicity, throughout this section we take the special compatible family (Φ, H, L) described in Section 1 as our basic CHNS. In particular, to our end we shall make the following additional assumption (H2): ∀t∈R+,v∈Φ−p,p≥p0, there exist positive constants ν > 0, µ > 0, p≤r≤qand positive function γ(t), t∈R+, such that 2A(t, v)[θqv] + kB(t, v)k2 L(2)(HQ,Φ−r)≤ −νkvk2 −r+γ(t)e−µt (3.1) where p0,qare introduced as in the assumption (H1) and γ(t) satisfies that for arbitrary δ > 0, γ(t) = o(eδt), as t→ ∞, i.e., limt→∞ γ(t)/eδt = 0. Before proceeding to our stability arguments, let us first make the following comments on the condition (H2): Remark 1. As is well known, the coercivity condition (2.11) plays an essential role in the establishment of the existence and uniqueness of the equation (2.8). The further restrictive coercivity condition (3.1) will play the role of an exponential stability criterion as described below. Remark 2. The exponential decay term appearing on the right hand side of (3.1) is of the essence for our stability purposes. In fact, to see this, let us simply consider the following one dimensional linear Itˆo equation: Example 3.1. Assume Xtsatisfies the following dXt=−pXtdt + (1 + t)−qdWt, t ≥0 with initial data X0= 0, where p,q > 0 are two positive constants and Wtis a onedimensional standard Brownian motion. Clearly, the left-hand side of the coercivity type condition (3.1) now turns out to be 2<−pv, v > +h(1 + t)−qi2=−2pv2+ (1 + t)−2q.(3.2) where <·,·>denotes the standard inner product in R. However, since the last term (1 + t)−2qis not exponentially decreasing, the solution is exponentially unstable. Indeed, it is easy to obtain the explicit solution Xt=e−pt Zt 0 eps ·(1 + s)−qdWs=: e−ptMt, t ≥0, which immediately implies that for arbitrarily given q > 0 Lyapunov exponent lim t→∞ log E|Xt|2 t= 0. 7 In the meantime, noticing the law of the iterated logarithm lim sup t→∞ Mt √2 log log = 1 a.s. and lim sup t→∞ log ³Rt 0e2ps(1 + s)−2qds´ t= 2p, we therefore get Lyapunov exponent lim sup t→∞ 1 tlog |Xt|= 0 a.s. That is, in spite of the typical stability of an ordinary differential equation dXt=−pXtdt, the polynomial type decay of the noise term is not sufficient to ensure the exponential stability of its stochastically perturbed system. Now we are in a position to obtain our main results in the paper. Theorem 3.2. Suppose Xtis a solution to the equation (2.8) satisfying (H1). Furthermore we assume the coercivity condition (3.1) holds. Then there exist constants τ > 0, C > 0such that EkXtk2 −r≤C·e−τt,∀t≥0.(3.3) That is, the strong solution is exponentially stable in mean square. In particular, constant τ > 0can be taken as follows: τ < µ, if µ≤νand τ=ν, if µ > ν. Proof. For arbitrary φ∈Φ, we have Xt[φ] = Zt 0 A(s, Xs)[φ]ds +X jZt 0 < B(s, Xs)0φ, vj>HQdWs[ı−1vj],(3.4) where {vj} ⊂ R(ı) is a CONS of HQand ıis defined as in Section 2. Here B(s, ·)0denotes the dual operator of B(s, ·)∈ L(HQ,Φ−r), s≥0. It follows from Itˆo’s formula and Definition 2.4 that for arbitrary δ > 0 with µ−δ > 0, we have e(µ−δ)t∧σnXt∧σn[φ]2−X0[φ]2 =(µ−δ)Zt∧σn 0 e(µ−δ)sXs[φ]2ds + 2 Zt∧σn 0 e(µ−δ)sXs[φ]A(s, Xs)[φ]ds + 2 X jZt∧σn 0 e(µ−δ)sXs[φ]HQdWs[ı−1vj] +Zt∧σn 0 e(µ−δ)sQ(B(s, Xs)0φ, B(s, Xs)0φ)ds 8 where (σn) is the sequence of stopping times defined as in Definition 2.4. Now, since Rt∧σn 0e(µ−δ)sXs[φ]HQdWt[ı−1vj], t∈R+, is a continuous martingale, it follows that E³Zt∧σn 0 e(µ−δ)sXs[φ]HQdWs[ı−1vj]´= 0, t ∈R+. Therefore, letting φ=φr k,n→ ∞,k∈Nand then adding on index k∈N, we can deduce by Fatou’s lemma and the condition (3.1) Ee(µ−δ)tkXtk2 −r ≤EkX0k2 −r+ (µ−δ−ν)Zt 0 e(µ−δ)sEkXsk2 −rds +Zt 0 γ(s)e−δsds. (3.5) If µ−ν≤0, we therefore deduce Ee(µ−δ)tkXtk2 −r≤EkX0k2 −r+Zt 0 γ(s)e−δsds, that is, letting k(δ) = R∞ 0γ(s)e−δsds, we have EkXtk2 −r≤³EkX0k2 −r+k(δ)´e−(µ−δ)t. On the other hand, if µ−ν > 0, it is always possible to choose a suitable δ > 0 such that µ−ν−δ > 0. Then, by virtue of Gronwall’s lemma we easily derive from (3.5) that Ee(µ−δ)tkXtk2 −r≤³EkX0k2 −r+Zt 0 γ(s)e−δsds´e(µ−δ−ν)t. Hence, letting δ > 0 small enough immediately yields that there exists a constant k(δ)>0 such that EkXtk2 −r≤³EkX0k2 −r+k(δ)´e−νt. Combining the arguments above, we thus obtain our conclusion. Theorem 3.3. Assume the assumptions in Theorem 3.2 hold. Then there exist positive constants M,βand a subset Ω0⊂Ωwith P(Ω0)=0such that, for each ω6∈ Ω0, there exists a positive random number T(ω)such that the following holds: kXtk2 −r≤M·e−βt,∀t≥T(ω).(3.6) That is, the strong solution is almost surely stable. Proof. Our proofs are divided into the following several steps. Step 1. We firstly claim that there exists a constant C > 0, independent of t∈R+, such that Zt s EkB(u, Xu)k2 L(2)(HQ,Φ−r)du ≤C < ∞,0≤s≤t. (3.7) 9