Comparison of the long-time behaviour of linear Ito and Stratonovich partial differential equations

comparison of the long-time behaviour of linear ito and stratonovich partial differential equations Tom´as CARABALLO and Jos´e A. LANGA Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Apdo. Correos 1160, 41080-SEVILLA (Spain). e-mails: [email protected] ; [email protected] To the memory of the first author´s father Abstract In this paper, we point out the different long-time behaviour of stochastic partial differential equations when one considers the stochastic term in the Ito or Stratonovich sense. In particular, we prove that the Stratonovich interpretation may not produce modification in the exponential stability of the deterministic model for a wide range of stochastic perturbations, while Ito’s one can give different results. In fact, some stabilization or destabilization effect can be obtained. Keywords: Almost sure exponential stability, stabilization, destabilization, linear Ito PDE, linear Stratonovich PDE. 1 1 Introduction and statement of the problem As it is well known, stochastic partial differential equations arise in the modelling of numerous problems from Physics, Biology, Chemistry,... It is assumed that the real phenomenon is better described if one considers a random or stochastic term in the equation of the model. This fact implies that we need to give a sense to the new nondeterministic equation. Two interpretation are the most commonly used in the literature: Ito’s stochastic equation and Stratonovich’s one. Each interpretation gives a different solution of the stochastic equation, so they provide different answers to the same problem. There exist several reasons which make reasonable both possibilities and there exists a rule which permit us to pass from one kind of equation to the other (see Arnold [1], Oksendal [10], Kunita [9],...). However, when one is analyzing the long-time behaviour of the solutions, special care should be paid to the choice of the model since the solutions of both stochastic equations can have totally different behaviour. This will be the main aim of this paper. Indeed, we are going to show that a linear deterministic model is exponentially stable if and only if when a certain class of Stratonovich noise is added to the problem, this remains exponentially stable. However, when the noise is considered in Ito’s sense, several different situations are possible, that is, it may happen that the deterministic and stochastic model are both exponentially stable, or that the deterministic is unstable and the stochastic is stable (stabilization), or even the former is stable while the latter unstable (destabilization). We would like to point out that we are going to restrict ourselves to the consideration of the linear case since, in this important situation, we will be able to prove necessary and sufficient conditions for the stability of the Stratonovich stochastic equation, and criteria for the Ito one, in terms of the stability of the corresponding deterministic one, and making use of simple arguments from the theory of linear semigroups, showing once again the power of this theory in dealing with stochastic problems. No doubt at all, our treatment admits extensions to semilinear and more general nonlinear situations, however, in these cases, only sufficient conditions can be obtained. Due to these reasons, we have preferred to consider the linear case described below. 2 Now, we are going to give the statement of the problem we shall study in this work. Let us consider the following deterministic evolution equation du(t) dt =Au(t), where Ais an unbounded linear operator in the real separable Hilbert space H(with norm |·|and inner product (·,·)), with domain D(A) dense in H, i.e. A:D(A)⊂H−→ H, and assume that Ais the generator of a c0-semigroup, S(t). Let u(t; 0, u0) denote the solution of the following problem (du(t) dt =Au(t) u(0) = u0∈H(1) It is well known that u(t; 0, u0) = S(t)u0,for u0∈D(A). Moreover, if S(t) is analytic, then u(t; 0, u0) = S(t)u0, for all u0∈H. Definition 1 Equation (1) (or the semigroup S(t)) is said to be exponentially stable, or simply stable, if there exist M≥1, γ > 0such that |S(t)| ≤ Me−γt,∀t≥0.(2) The following result is due to Datko [5]. Theorem 2 The following statements are equivalent: i) S(t)is stable; ii) R∞ 0|S(t)x|dt < ∞, for each, x∈H; iii) There exists a self-adjoint nonnegative operator P∈ L(H)such that 2(Ax, Px) = −(x, x),for each x∈D(A). Let us now consider that a linear noise is added to the problem (1). Thus, we can consider, respectively, the Stratonovich and Ito versions of the stochastic equation: 3 (du(t) = Au(t)dt +Bu(t)◦dwt u(0) = u0∈H, (3) (du(t) = Au(t)dt +Bu(t)dwt u(0) = u0∈H, (4) where, for the sake of simplicity, we assume that wtis a real standard Wiener process, defined on the filtered and complete probability space (Ω,F,(Ft)t≥0,P), and Bis a linear operator (bounded or not) in H. We want to mention that, although our analysis can be extended to the more general situation of considering a sum of the form Pd i=1 Biudwi tinstead of Budwt, where wi tare mutually independent standard Wiener processes, we prefer to develop this case for the sake of clarity. As we are interested in the stability analysis of the solutions to equations (3) and (4), we assume that, for each u0∈H, there exists a unique strong (and therefore, mild) solution of both problems (see Da Prato-Zabczyk [4] and Kunita [9] for the definitions, properties and conditions under which this holds). 2 The stability of Stratonovich equations Let u(t, ω; 0, u0) denote the unique solution to the problem (du(t) = Au(t)dt +Bu(t)◦dwt u(0) = u0∈H, (5) where we assume that the operator B:D(B)⊂H→His the generator of a c0−group, denoted SB(t),satisfying D(A)⊂D(B). We now recall the following definition. Definition 3 The zero solution of problem (5) is said to be exponentially asymptotically stable with probability one (w.p.1) if there exist N⊂Ω,P(N) = 0,and α, β > 0,such that for every ω /∈N, there exists T(ω)>0such that |u(t, ω; 0, u0)| ≤ β|u0|exp(−αt),∀t≥T(ω), u0∈D(A) 4 The following result holds Theorem 4 Assume that Aand SB(t)commute.Then, u≡0is exponentially stable as solution of Eq. (1) if and only if u≡0is exponentially asymptotically stable w.p.1 as solution of (5). Proof. Let us make the following change: z(t) := z(t, ω; 0, u0) = S−1 B(wt(ω))u(t, ω; 0, u0).(6) Now, it is not difficult to check that dz(t) = S−1 B(wt(ω))du(t)−S−1 B(wt(ω))Bu(t)◦dwt =SB(−wt(ω))Au(t)dt =Az(t)dt, and z(0) = u0.Consequently, the process z(t) is solution to Eq. (1) P−a.s., in fact z(t) = S(t)u0,and the solutions to (1) and (5) are related by means of (6). Now, we can prove the theorem. Assume that problem (1) is exponentially stable. This means that there exist M≥1, γ > 0 such that |S(t)| ≤ Me−γt,for all t≥0.On the other hand, as the operator Bis the generator of a c0−group, it is well known (see Pazy [11]) that there exist σ∈R, b ≥1,such that |SB(t)| ≤ beσ|t|,for all t∈R. Then, since P−a.s. lim t→+∞ |wt(ω)| t= 0,(7) there exists N⊂Ω,P(N) = 0 such that if ω /∈N, then lim t→+∞(γ−σ|wt(ω)| t) = γ, and, there exists T(ω) such that for all t≥T(ω) γ−σ|wt(ω)| t≥γ 2. Thus, given u0∈D(A), ω /∈N, and taking into account that z(t) = S(t)u0, |u(t, ω; 0, u0)|=|SB(wt(ω))z(t)| ≤Mbeσ|wt(ω)|e−γt|u0| ≤Mb|u0|e−(γ−σ|wt(ω)| t)t ≤Mb|u0|e−γ0t,∀t≥T(ω), 5 where γ0=γ/2.Therefore, u≡0 is exponentially asymptotically stable w.p.1 as solution of Eq. (5). Conversely, if Bis an operator such that u≡0 is exponentially asymptotically stable w.p.1 as solution of Eq. (5), there exist N0⊂Ω,P(N0) = 0, and α, β > 0, such that if ω /∈N0, there exists T0(ω)>0 such that |u(t, ω; 0, u0)| ≤ β|u0|e−αt,∀t≥T0(ω) Now, for a fixed ω /∈N0,(6) implies |S(t)u0|≤|SB(−wt(ω))u(t, ω; 0, u0)|, ≤bβ|u0|e−(α−σ|wt(ω)| t)t,∀t≥T0(ω). On the other hand, from (7) we can assure the existence of N1⊂Ω, P(N1) = 0, such that for ω /∈N1, there exists T1(ω)>0 such that for all t≥T1(ω),it holds α−σ|wt(ω)| t≥α 2. Denoting N=N0SN1and taking a fixed ω /∈N, it easily follows that |S(t)u0| ≤ Me−γt|u0|,∀t≥e T, where e T=] T(ω) = max{T0(ω), T1(ω)}. The proof is now complete. Remark 5 Notice that what we have proved is: i) If the deterministic system (1) is exponentially stable, then for all linear stochastic perturbations (in the Stratonovich sense) of the kind considered in Theorem 4, the perturbed system remains exponentially stable. ii) Conversely, if there exists a perturbation of the mentioned kind that makes the stochastic system become exponentially asymptotically stable, then the deterministic system must be exponentially stable too. In conclusion, we can afirm that stochastic perturbations of this type do not modify the long-time behaviour of the solutions of the deterministic problem. 6 Remark 6 a) Observe that if we consider the particular case Bu =σu, for some σ∈R,and all u∈H, the c0−group SB(t)is given by SB(t) = eσtI, and the hypotheses in Theorem 4 are fulfilled. Therefore, the null solution of (1) is exponentially stable iff the null solution to Eq. (5) is exponentially asymptotically stable w.p.1 for all σ∈R. b) It is worth mentioning that, in the finite dimensional case, Arnold [2] proves that the deterministic system dx(t) = Ax(t)dt can be stabilized by a suitable Stratonovich linear noise if and only if trace A < 0. However, as far as we know, a similar result in infinite dimension remains as an open question and it seems difficult to develop a similar analysis to the one in Arnold [2]. Nevertheless, if we consider stochastic perturbations in Ito’s sense much more can be obtained as we are going to show in the next Section. 3 The stability of Ito equations In this Section, we will prove that the long-time behaviour of the deterministic problem can be modified if we consider the stochastic perturbation in the Ito sense. Let us now consider the Ito equation (du(t) = Au(t)dt +Bu(t)dwt u(0) = u0∈H. (8) It is well known that this is equivalent to the following Stratonovich formulation (see Kunita [9]) (du(t) = (Au(t)−1 2B2u(t))dt +Bu(t)◦dwt u(0) = u0∈H, (9) where we assume that the linear operator B:D(B)⊂H→Hsatisfies D(A)⊂D(B2) and generates a c0−group. Let us denote C=A−1 2B2and assume that Cis the generator of a c0−semigroup SC(t), which automatically holds if, for instance, B∈ L(H) (see, Pazy [11] or Curtain and Pritchard [3]). By the virtue of the analysis in the preceding Section, we can assure that, under the additional hypothesis of commuting Aand SB(t), Eq. (8) is 7 exponentially stable (or, equivalently, (9) is exponentially stable) if and only if SC(t) is stable. Consequently, the following three cases are possible: 1. S(t) is stable and the operator Bis such that SC(t) is also stable (the stability of the deterministic system remains under the stochastic perturbation) 2. S(t) is not stable but the operator Bmakes SC(t) be stable (a stabilization effect has been produced by the random perturbation) 3. S(t) is stable but SC(t) is not stable (the deterministic system has been destabilized by the noise) Let us now prove some result concerning these possibilities in the following subsections. 3.1 Case 1: stability results We can first prove the following result concerning stability. Theorem 7 a) Assume S(t)is stable (i.e. it holds (2)) and B∈ L(H)generates a c0−group. Then, the semigroup SC(t)is stable if |B|2 L(H)<2γ M. b) In particular, if in addition to the hypotheses in a), the operator Bis defined as B(u) = bu, for all u∈H, where b∈R, then, the semigroup SC(t)is stable whatever bbe. Proof. a) It is known (see Curtain-Pritchard [3], Theorem 10.9 page 210) that SC(t) satisfies |SC(t)|L(H)≤Me(−γ+M 2|B|2 L(H))t. Thus, if −γ+M 2|B|2 L(H)<0, SC(t) is stable. 8 b) In the particular case B(u) = bu, we have that SC(t) = e−b2 2tS(t).So, we easily get that |SC(t)| ≤ Me−(γ+b2 2)t,and for all b∈Rit follows that γ+b2 2>0. We immediately can prove the following consequence. Corollary 8 Assume that the semigroup S(t)is stable and the operator A commutes with the c0−group generated by B∈ L(H).Then, Eq.(8) is stable provided |B|2 L(H)<2γ M. Proof. It follows from a) in the preceding theorem and the equivalence between the stability of semigroup SC(t), Eq. (9) and (8). Remark 9 Notice that Theorem 7 gives a different and simpler proof of some results in Haussmann [6] and Ichikawa [7] in the particular case of commuting operators Aand SB(t)(the group generated by B). Moreover, in the case considered in part b) our results improves theirs since we get stability for all b∈R,while they only can assure exponential stability for small values of b. 3.2 Case 2: stabilization results Firstly, we are going to prove that when the semigroup S(t) is not stable, we can always choose a suitable linear operator Bsuch that SC(t) becomes stable. Theorem 10 Assume Agenerates a c0−semigroup S(t)and consider the linear operator Bdefined as Bu =bu, for some b∈R. Then, the semigroup SC(t)generated by C=A−1 2B2is stable provided bis large enough. Proof. Indeed, it is known that there exists M≥1 and ρ∈Rsuch that |S(t)| ≤ Meρt,∀t≥0. Thus, by similar computations as in part b) in Theorem 7, we can easily get that |SC(t)| ≤ Me(ρ−b2 2)t,∀t≥0,and taking b large enough, we obtain that ρ−b2 2<0,and SCis therefore stable. However, it is not necessary to set B(u) = bu in the theorem in order to get stabilization, as can be deduced from the next more general result. 9