Stochastic Differential Equations in Infinite Dimensions

The study of infinite–dimensional equations is motivated by applications to partial differential equations. We describe two methods used for deterministic equations. One involves the study of mild solutions with the help of a semigroup generated by the unbounded operator on a Banach space, as in the work of Pazy (Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York 1983). We illustrate this approach in an example of the heat equation cast as an abstract Cauchy problem, where the state space of the solution is an infinite-dimensional Banach space, and the coefficient, even in this linear case, is an unbounded operator. We present here a comprehensive review of the properties of semigroups of operators. In the other method, the unbounded operator is recast as a bounded operator from its domain Hilbert space V to its range V ^∗, the continuous dual of V, where the equation is defined. However, the initial condition is in a Hilbert space H, with V↪H↪V ^∗, where the embeddings are dense and continuous. This approach requires a condition of coercivity to bring the solution at time t>0 into H, the space of the initial condition. We present here the approach of Lions (Équations Différentielles Opérationelles et Problèmes aux Limites. Springer, Berlin, 1961), to show that the solution is a continuous H-valued function on [0,T]. In the presence of non-linear coefficients, the solution may not exist due to the failure of the Peano theorem (since the Arzela–Ascoli theorem is invalid in infinite dimensions), and to secure the existence result we will assume in later chapters compactness of the embeddings or of the semigroup of operators.

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, 1974) and by Øksendal (Stochastic Differential Equations. Springer, New York, 1998). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

There exist different notions of a solution to a semilinear stochastic differential equation (SSDE). We define strong, weak (in the sense of duality), mild, and martingale solutions, and study the problem of existence and uniqueness. As in the deterministic case, for example in Pazy (Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York 1983), we first study solutions to a stochastic counterpart of the deterministic inhomogeneous Cauchy problem and we highlight the role played by the stochastic convolution. The SSDE’s we investigate are allowed to depend on the entire past of the solution which significantly broadens the field of applications. The existence result for mild solutions is first obtained for equations with Lipschitz coefficients. In the special case of equations depending only on the presence, we discuss the Markov property, dependence of the solution on the initial condition, including differentiability, and the Kolmogorov backward equation. We also study SSDE’s with continuous coefficients, and present an existence result for martingale solutions, but due to the failure of the Peano theorem, a compactness assumption is added for the associated semigroup, as in DaPrato and Zabczyk (Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge 1992). We also present an existence result for SSDE’s driven by a cylindrical Wiener process.

In case of SDE’s with continuous coefficients, we use Lipschitz-type approximation, as in the work of Gikhman and Skorokhod (The Theory of Stochastic Processes. Springer, Brelin, 1974), and prove the existence of weak solutions (in the stochastic sense). However, again due to the failure of the Peano theorem, the solution is in a larger Hilbert space, where the space containing the initial condition is compactly embedded. We call this technique the “method of compact embedding”, and use it later for finding solutions in the variational method and for models of spin systems.

The variational method for solving stochastic partial differential equations (SPDE’s) of evolutionary type involves recasting them as SDE’s in a Gelfand triplet of Hilbert or Banach spaces V↪H↪V ^∗, where the embeddings are dense and continuous. We discuss only the case of separable Hilbert spaces. In order to construct a weak solution, we assume that the embeddings are compact, and use the “method of compact embedding” introduced in Chap. 3, together with the stochastic analogue of Lions’ theorem from Chap. 1. The solution is an H-valued stochastic process with continuous sample paths. Under the assumption of monotonicity, we obtain unique strong solution using pathwise uniqueness.

We also present the result on the existence of strong solutions, following the ideas in Prévôt and Röckner (A Concise Course on Stochastic Partial Differential Equations. LNM, vol. 1905. Springer, Berlin, 2007). Assuming that the coefficients are monotone suffices to produce a strong solution without the need for compactness of the embeddings in the Gelfand triplet. Using again the stochastic analogue of Lions’ theorem allows to put the solution in H and assure continuity of its sample paths. We also present results on Markov and strong Markov properties of strong variational solutions.

The study of interacting particle systems arise in physics in case of spin systems and behavior of systems following Glauber dynamics. They can be modeled with stochastic differential equations in ℝ^∞ or in \({\mathbb{R}}^{{\mathbb{Z}}^{d}}\), d>1. One wishes to know if a solution exists and determine its state space. For example, it may be interesting if the solution is in l ₂. However, we consider models where the drift coefficient is not continuous on l ₂. This complication, together with the fact that the Peano theorem fails in infinite dimensions, can be overcome by noting that the embedding from l ₂ to ℝ^∞ is continuous and compact, and by applying the “method of compact embedding” from Chap. 3. We construct weak solutions, generally in a Hilbert space H, which are not continuous as functions into the state space with its natural topology. However, by identifying H with l ₂↪ℝ^∞, they turn out to be continuous in the topology induced on H from ℝ^∞. This part is related to the work of Leha and Ritter (Math. Ann. 270:109–123, 1985), where equations in general form are studied and the motivation comes from modeling of unbounded spin systems. We show the existence of solutions under weaker condition on the drift and provide Galerkin approximation in the case studied by Leha and Ritter. In the second part we prove existence of solutions for quantum lattice systems in \({\mathbb{R}}^{{\mathbb{Z}}^{d}}\) under weaker assumptions than those considered by Albeverio et al. (Rev. Math. Phys. 13(1):51–124, 2001) in justifying Glauber dynamics. To that end, we change the set-up to equations in a dual to a nuclear space and obtain weak solutions using compact embeddings.

If is a finite-dimensional Hilbert space, then Lyapunov proved that the mild solution {u ^x (t)} of the Cauchy problem \(\dot{u} (t) = Au(t)\), with the initial condition satisfies the following condition (a) , with \(r,\;c_{0}>0\), for t>0, if and only if, there exists a positive definite matrix R satisfying two conditions: (i) , , \(c_{1}\; c_{2}>0\), and (ii) A ^∗ R+RA=−I. The infinite dimensional analogue of this theorem is given by Datko. However, in this case the operator R does not satisfy the lower bound in condition (i). Lyapunov uses this lower bound crucially in going from linear to non-linear case. The solution in non-linear case will satisfy condition (a), termed exponential stability. When the operator A generates a pseudo-contraction semigroup, we produce an operator R satisfying both conditions (i) and (ii) in a deterministic equation. At this stage, we discuss stochastic equations and demonstrate how to study exponential stability for mild solutions of non-linear equations. We cannot use the Itô formula for mild solutions of stochastic differential equations, thus we have to approximate them using Yosida approximation, which produces a sequence of strong solutions. We use this approximation to obtain a sufficient condition for exponential stability in the mean square sense (m.s.s.) when the Lyapunov function Ψ exists. However, in order to use this approximation, we need to assume that Ψ is locally bounded. In the linear case we produce such a Lyapunov function obtaining necessary and sufficient conditions for the exponential stability in the m.s.s. of the solution. Then the first order approximation is used to study the exponential stability in the m.s.s. of mild solutions to non-linear equations following the ideas of Lyapunov. We provide the stochastic analogue of Datko’s theorem in the Appendix.

We also present Lyapunov function approach for studying exponential stability in the m.s.s. of strong variational solutions in a Gelfand triplet of real separable Hilbert spaces V↪H↪V ^∗. In the deterministic linear case, A:V→V ^∗ is a bounded operator satisfying the coercivity and monotonicity conditions. Then the solution of the Cauchy problem u ^x (t) is in C([0,T],H) We prove that u ^x (t) is exponentially stable if and only if there exists an operator \({\tilde{C}}\) satisfying conditions (i) and (ii). However the norm in V is used to define \({\tilde{C}}\) through the equality \(\langle {\tilde{C}}x,x\rangle_{H} = \int_{0}^{\infty}\| u^{x}(t)\|^{2}_{V}\, dt\). In order to show that \(\langle {\tilde{C}}x,x\rangle_{H}\) is finite we need the coercivity condition. We follow the same plan as for mild solutions for stochastic equations. We obtain a sufficient condition for exponential stability in the m.s.s. in the non-linear case in terms of the existence of the Lyapunov function. For linear equations we obtain necessary and sufficient conditions for exponential stability in the m.s.s. by constructing the Lyapunov function. We extend that result to non-linear equations using the first order approximation of non-linearities.

We introduce the concept of stability in probability for solutions to stochastic equations and provide a sufficient condition in terms of the existence of the Lyapunov function. Using the previous results, we show that exponential stability in the m.s.s. implies stability in probability.

We study ultimate boundedness in the m.s.s. of solutions to SDE’s, namely the following property, \(E\| X^{x}(t)\|^{2}_{H}\leq c\mathrm{e}^{-\beta t}\| x\|_{H}^{2}+M\), \(c,\; \beta >0\), M<∞.

The methods used here are similar to those employed in the study of exponential stability in the m.s.s. and involve obtaining necessary and sufficient conditions for ultimate boundedness in the m.s.s. in terms of the Lyapunov function. We achieve this goal both for mild and for strong variational solutions. As a consequence we obtain sufficient conditions for ultimate boundedness in the m.s.s. in terms of exponential stability of a solution of the related deterministic problem. We also obtain conditions for non-linear equations in terms of the behavior of their coefficients at infinity (in norm). Our results are applied to study the behavior of solutions for specific examples of SPDE’s.

Next, we show that under the assumption of ultimate boundedness in the m.s.s. there exists an invariant measure for mild solutions and for strong variational solutions in a Gelfand triplet V↪H↪V ^∗, provided the embeddings are compact. Under this compactness assumption we prove the recurrance of strong variational solutions to a compact set. Finally, we examine asymptotic behavior of solutions to SPDE’s such as the heat equation, Navier–Stokes equation, and of mild solutions to semilinear equations driven by a cylindrical Wiener process.