On the Pathwise Exponential Stability of Nonlinear Stochastic Partial-Differential Equations

ON THE PATHWISE EXPONENTIAL STABILITY OF NON–LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Tom´as Caraballo and Jos´e Real Dpto. Ecuaciones Diferenciales y An´alisis Num´erico. Universidad de Sevilla. Apartado de Correos 1160. 41080–SEVILLA (Spain) December 15, 2009 Abstract Sufficient conditions to get exponential stability for the sample paths (with probability one) of a non-linear monotone stochastic Partial Differential Equation are proved. In fact, we improve a stability criterion established in Chow [3] since, under the same hypotheses, we get pathwise exponential stability instead of stability of sample paths. 1INTRODUCTION AND PRELIMINARIES The aim of this paper is to give sufficient conditions in order to get exponential stability for the sample paths of certain stochastic PDE of evolution type. In fact, we consider the following equation dxt=A(t, xt)dt +B(t, xt)dwt, t > 0 x0=h(1) where A(t, ·) and B(t, ·) are families of (non-linear) operators in Hilbert spaces and wtis a Hilbert Wiener process. This equation has been studied by several authors over the last years. For instance, Pardoux [7], Ichikawa [6] and Caraballo [1] (among others) have established results on the existence and uniqueness of solutions. However, we are now interested in the analysis of the exponential stability for the paths of the trivial solution of (1). We can mention here that Haussmann [5] obtained pathwise exponential stability with probability one (w.p.1) for linear Aand Band, Caraballo [2] generalized these results to the delayed case. Also, Ichikawa [6] proved similar results for the mild solution of the semilinear case, i.e. for linear Aand Lipschitz continuous B. At the same time, Chow [3] obtained asymptotic stability for the sample paths of (1) when Aand Bdo not depend on tand a coercivity condition holds. Recently, Chow and Menaldi [4] have obtained some estimates in exit probability for the strong solution of the semilinear problem. Nevertheless, we still have not found in the literature the study of exponential stability of paths in a more general case, i.e. for non–linear monotone Aand Lipschitz continuous B. So, we are going to analyze it in this paper. One of our results improves Theorem 5.2 from Chow [3] since, under the same hypotheses, we get exponential stability for the paths instead of the asymptotic stability that he obtains. First, we give sufficient conditions for the exponential stability in mean square of the trivial solution of (1). Next, we obtain exponential stability of paths (w.p.1). Our method is based in appying Itˆo’s formula for a suitable function and using a coercivity condition. Consequently, we will observe how the coercivity condition may be regarded as a exponential stability criterion. Now we are going to state our problem in a suitable form: Let Vbe a Banach space and H, K real, separable Hilbert spaces such that V ֒→H≡H′֒→V′, where the injections are continuous and dense. We denote by k · k ,| · | and k · k∗the norms in V,Hand V′respectively; by h·,·i the duality product between V′, V , and by (·,·) the scalar product in H. Let wtbe a Wiener process defined on the complete probability space (Ω,F, P) and taking values in the separable Hilbert space K, with incremental covariance operator W. Let (Ft)t≥0be the σ-algebra generated by {ws,0≤s≤t}, then wtis a martingale relative to (Ft)t≥0and we have the following representation of wt: wt= ∞ X i=1 βi tei, where (ei) is an orthonormal set of eigenvectors of W,βi tare mutually independent real Wiener processes with incremental covariance λi>0, Wei=λieiand tr W= P∞ i=1 λi(tr denotes the trace of an operator, see Pardoux [7]). As an abuse of notation, we also use |·| for the norm in the linear continuous operator space L(K, H). We denote by Ip(0, T;V),for p > 1 and T > 0,the space of V–valued processes (xt)t∈[0,T ](we will write xtfor short) measurable (from [0, T]×Ω in V), and satisfying: 1. x(t) is Ft−measurable a.e. in t(in the sequel, we will write a.e.t.) 2. ERT 0|xt|pdt < +∞. It is not difficult to check that the space Ip(0, T;V) is a closed subspace of Lp(Ω ×[0, T],F ⊗ B([0, T]), dP ⊗dt;V),where by B([0, T]) we denote the Borel σ–algebra. For short, we shall write L2(Ω; C(0, T;H)) instead of L2(Ω,F, dP;C(0, T;H)) , where C(0, T;H) denotes the space of continuous functions from [0, T] to H. Let A(t, ·) : V→V′be a family of non linear operators defined a.e.t. satisfying A(t, 0) = 0 for all t≥0 , and let p > 1.(Note that we assume A(t, 0) = 0 because we are going to restrict ourselves to the stability analysis for the trivial solution of (1)). We also consider a family of operators B(t, ·) : V→ L(K, H) defined a.e.t., and satisfying: (b.1) B(t, 0) = 0 ,∀t≥0 (b.2) Lipschitz condition: There exists k1such that |B(t, x)−B(t, y)| ≤ k1kx−yk,∀x, y ∈V , a.e.t. (b.3) Measurability: t∈(0, T)→B(t, x)∈ L(K, H) is Lebesgue–measurable ∀x∈V , ∀T > 0. Since we are mainly interested in exponential stability questions, we will assume there exists a unique process x∈Ip(0, T;V)∩L2(Ω; C(0, T;H)) ,∀T > 0, which is solution of (1). In other words, xtverifies the following integral equation in V′: xt=x0+Zt 0 A(s, xs)ds +Zt 0 B(s, xs)dws, t > 0, P −a.s.(2) where x0=h∈L2(Ω,F0, P;H). Observe that, we can assure existence and uniqueness of solution for equation (2) if, for instance, the following conditions hold (see Pardoux [7]) (a.1) Coercivity: There exist α > 0 and λ, γ ∈such that: −2hA(t, x), xi+λ|x|2+γ≥αkxkp+kB(t, x)W1/2k2,∀x∈V , a.e.t. , where k · k2denotes the Hilbert–Schmidt norm of nuclear operators, i.e. kB(t, x)W1/2k2= tr (B(t, x)WB(t, x)∗) (a.2) Monotonicity: −2hA(t, x)−A(t, y), x −yi+λ|x−y|2≥ k(B(t, x)−B(t, y))W1/2k2, forallx, y ∈ V, a.e.t. (a.3) Boundedness: There exists c > 0 : kA(t, x)k∗≤ckxkp−1,∀x∈V , a.e.t. (a.4) Hemicontinuity: The map θ∈ → hA(t, x +θy), zi ∈ is continuous ∀x, y, z ∈V , a.e.t. (a.5) Measurability: t∈(0, T)→A(t, x)∈V′is Lebesgue −measurable ∀x∈V , a.e.t., ∀T > 0. 2THE MAIN RESULTS We note that there exists a positive constant βsuch that |x| ≤ βkxk ∀x∈V. Now, we state the exponential stability for the second moment of xt, solution of (2). Theorem 2.1 Assume conditions (b.1)–(b.3) and (a.1). Then, there exists r > 0 such that E|xt|2≤E|x0|2e−rt ∀t≥0,(3) if either one of the following hypotheses holds: (a) λ < 0, γ ≤0 (∀p > 1) (b) λβ2−α < 0, γ ≤0 (p= 2). Proof. We apply Itˆo’s formula (see Pardoux [7], Caraballo [1]) for the function eαt| · |2and the process xt, where r > 0 is such that r+λ < 0 in case (a) or (r+λ)β2−α < 0 in case (b) (we note that there exists such rsince the maps r→r+λ , r →(r+λ)β2−αare continuous and (a) or (b) holds). We then obtain ert|xt|2− |x0|2=rZt 0 ers|xs|2ds + 2 Zt 0 ershxs, A(s, xs)ids +2 Zt 0 ershxs, B(s, xs)dwsi +Zt 0 erstr (B(s, xs)WB(s, xs)∗)ds. (4) Now, since R· 0ershxs, B(s, xs)dwsiis a continuous local martingale (see Caraballo [1] and Chow [3]), it follows EZt 0 ershxs, B(s, xs)dwsi= 0 . Therefore, from (a.1) and (4) we can deduce ertE|xt|2≤E|x0|2+ (r+λ)Zt 0 ersE|xs|2ds −αZt 0 ersEkxskpds +γ ert −1 r!.(5) Now, if (a) holds, (5) yields ertE|xt|2≤E|x0|2, and if we suppose (b), (5) implies ertE|xt|2≤E|x0|2+h(r+λ)β2−αiZt 0 ersEkxsk2ds ≤E|x0|2. So, the proof is complete. Now, we are going to establish that the exponential stability of the second moment implies the exponential stability of the sample paths w.p.1. First, we need the following result. Lemma 2.1 Assume the solution xtof problem (2) satisfies (3). Then, there exist positive constants c1, c2such that (a) Rt τE|B(s, xs)|2ds ≤c1E|x0|2e−rτ ,0≤τ≤t (b) E sup 0≤t<+∞ |xt|2!≤c2E|x0|2. Proof. First, applying Itˆo’s formula as in Theorem 2.1 we get ertE|xt|2≤E|x0|2+h(λ+r)β2−αiZt 0 ersEkxsk2ds , ∀t≥0.(6) Since (λ+r)β2−α < 0, it follows Zt 0 ersEkxsk2ds ≤1 α−(λ+r)β2E|x0|2,∀t≥0. Now, (6) and (b.2) yield Zt 0 ersE|B(s, xs)|2ds ≤k1 α−(λ+r)β2E|x0|2≡c1E|x0|2,∀t≥0 and, for 0 ≤τ≤t, we get Zt τ ersE|B(s, xs)|2ds ≤c1E|x0|2 and e−rτ Zt τ ersE|B(s, xs)|2ds ≤c1E|x0|2e−rτ ,0≤τ≤t . Therefore Zt τ E|B(s, xs)|2ds ≤Zt τ er(s−τ)E|B(s, xs)|2ds ≤c1E|x0|2e−rτ ,0≤τ≤t , and hence (a) is proved. Next, Itˆo’s formula for |xt|2and the coercivity condition imply |xt|2=|x0|2+Zt 0 hxs, A(s, xs)ids + 2 Zt 0 hxs, B(s, xs)dwsi +Zt 0 tr (B(s, xs)WB(s, xs)∗)ds ≤ |x0|2+|λ|Zt 0 |xs|2ds + 2 Zt 0 (xs, B(s, xs), dws) , and so, E"sup t∈[0,T] |xt|2#≤E|x0|2+|λ|ZT 0 E|xt|2dt +2E"sup t∈[0,T]Zt 0 (xs, B(s, xs)dws)#.(7) Now, we estimate the terms on the right–hand side of (7). |λ|ZT 0 E|xs|2ds ≤ |λ|ZT 0 e−rsE|x0|2ds ≤|λ| rE|x0|2,∀T > 0.(8) Using Burkholder–Davis–Gundy’s inequality we obtain: E"sup t∈[0,T]Zt 0 (xs, B(s, xs)dws)#≤k3E  tr (W)ZT 0 |xs|2|B(s, xs)|2ds!1/2  ≤k4E  ZT 0 |xs|2|B(s, xs)|2ds!1/2  ≤k5E sup s∈[0,T ] |xs|"ZT 0 |B(s, xs)|2ds#1/2  ≤1 2E"sup t∈[0,T] |xt|2#+k6ZT 0 E|B(s, xs)|2ds ≤1 2E"sup t∈[0,T] |xt|2#+k7E|x0|2,(9) since (a) holds. Finally, as k7does not depend on T, we can obtain (b) from (7) −(9) and Lebesgue’s theorem. Theorem 2.2 Assume the hypotheses in Theorem 2.1. Then, there exist positive constants ξ, η and a subset Λ⊂Ωwith P(Λ) = 0 such that, for each ω6∈ Λ, there exists a positive number T(ω)such that the following holds: |xt(ω)|2≤ηE|x0|2e−ξt ∀t≥T(ω).(10) Proof. We only sketch the proof because it is similar to the linear case one (see Haussmann [5] and Caraballo [2], for details). First, we apply Itˆo’s formula. As γ≤0 and coercivity holds, we obtain: |xt|2≤ |xN|2+|λ|Zt N |xs|2ds + 2 Zt N (xs, B(s, xs)dws) (11) for t≥N , where Nis a natural number. Next, if INdenotes the interval [N, N + 1] we have: sup t∈IN |xt|2≤ |xN|2+|λ|ZN+1 N |xs|2ds +2 sup t∈INZt N (xs, B(s, xs)dws) and so, P"sup t∈IN |xt|2≥ε2 N#≤Ph|xN|2≥ε2 N/3i+P"|λ|ZN+1 N |xs|2ds ≥ε2 N/3# +P"2 sup t∈INZt N (xs, B(s, xs)dws) ≥ε2 N/3#,(12) where εN=E|x0|e−rN/4. Now, we can estimate the terms on the right–hand side of (12) using Kolmogorov’s inequality and (3) for the first two terms and we also use the inequalities of Burkholder–Davis–Gundy and Holder and Lemma 2.1 for the last. We then obtain P"sup t∈IN |xt|2≥ε2 N#≤k8e−rN/4,(13) and finally, Borel–Cantelli’s lemma completes the proof. Remark. We can observe how the coercivity condition (with (a) or (b)) implies pathwise exponential stability for the solutions of problem (2). However, Chow [3] only obtains asymptotic stability under the same hypotheses. We also note that assumption (iii) in [3, Theorem 5.2] never holds because from (a.1) with x= 0 it follows γ≥0. Consequently, in (a) and (b) we can set γ= 0 instead of γ≤0 and the same in Chow’s Theorem 5.2. 3EXAMPLES First, we note that, in the examples given in Chow [3], the trivial solution has exponentially stable paths instead of the asymptotic stability property stated there. Next, we are going to give a different example. Let Obe an open, bounded subset in Nwith regular boundary and let 2< p < +∞. We consider the Sobolev space V=W1,p 0(O) , H=L2(O) with their usual inner products, and the monotone operator A:V7→ V′defined as hAu, vi=− N X i=1 ZO ∂u(x) ∂xi p−2∂u(x) ∂xi ∂v(x) ∂xi dx −ZO a(x)u(x)v(x)dx ∀u, v ∈V , where a∈L∞(O) satisfies a(x)≥ˆa > 0,a.e. x ∈ O .We also consider B(u) = g(u), u ∈Vwhere g:7→ is Lipschitz continuous with constant Lsuch that L2< 2ˆaand g(0) = 0 . Finally, let wtbe a standard real Wiener process (so, K= and W= 1 ). Then, although condition (a.3) does not hold when kuk ≤ 1 , we can assure there exists a unique solution of (2), for each u0∈Lp(Ω,F0, P ;H) (see Pardoux [7], Theorem 4.1, p. 126). In this case, (a.1) holds with γ= 0 , λ = −ε < 0, p > 2, α = 2 ,where ε > 0 is such that L2≤2ˆa−ε . Consequently, we get asymptotic exponential stability for the paths of the trivial solution, u , of a problem which can be interpreted as follows:                du(t, x) = N X i=1 ∂ ∂xi  ∂u(t, x) ∂xi p−2∂u(t, x) ∂xi!−a(x)u(t, x)!dt +g(u(t, x)) dwt,a.e.in (0,+∞)× O u(0, x) = u0(x),a.e.in O u(t, x) = 0 ,a.e.in (0,+∞)×∂O References [1] T. Caraballo, Existence and Uniqueness of Solutions for Non–Linear Stochastic Partial Differential Equations, Collect. Math. 42, 1 (1991), 51–74. [2] T. Caraballo, Asymptotic Exponential Stability of Stochastic Partial Differential Equations with Delay, Stochastics and Stochastics Reports, 33 (1990), 27–47. [3] P. L. Chow, Stability of Nonlinear Stochastic–Evolution Equation, J. Math. Anal. Appl., 89 (2) (1982), 400–419. [4] P. L. Chow and J. L. Menaldi, Exponential Estimates in Exit Probability for some Diffusion Process in Hilbert Spaces, Stochastics and Stochastics Reports, 29 (1990), 377–393. [5] U. G. Haussmann, Asymptotic Stability of the Linear Itˆo Equation in Infinite– Dimension, J. Math. Anal. Appl., 65 (1978), 219–235. [6] A. Ichikawa, Stability of Semilinear Stochastic Evolution Equations, J. Math. Anal. Appl., 90 (1982), 12–44. [7] E. Pardoux, Equations aux D´eriv´ees Partielles Stochastiques Nonlin´eaires Monotones, Thesis, Universit´e Paris Sud, 1975.