A solution theory for a general class of SPDEs

Stochastics and Partial Differential Equations: Analysis and Computations, Dec 2016

In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of \({\mathbb {R}^d}\). The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green’s function is not required.

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A solution theory for a general class of SPDEs

Stoch PDE: Anal Comp A solution theory for a general class of SPDEs André Süß 0 1 Marcus Waurick 0 1 B Marcus Waurick 0 1 Zurich, Switzerland 0 Department of Mathematical Sciences, University of Bath , Claverton Down, Bath BA2 7AY , UK 1 Mathematics Subject Classification Primary 60H15 In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of Rd . The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green's function is not required. Stochastic partial differential equations; Evolutionary equations; Stochastic equations of mathematical physics; Weak solutions 1 Introduction The study of stochastic partial differential equations (SPDEs) attracted a lot of interest in the recent years, with a wide range of equations already been investigated. A common theme in the study of these equations is to attack the problem of existence and uniqueness of solutions to SPDEs by taking solution approaches from the deterministic setting of PDEs and applying them to a setting that involves a stochastic perturbation. Examples for this are the random-field approach that uses the fundamental solution to the associated PDE in [5,7,39], the semi-group approach which treats evolution equations in Hilbert/Banach spaces via the semi-group generated by the differential operator of the associated PDE, see [8] or [20,21] for a treatise, and the variational approach which involves evaluating the SPDE against test functions, which corresponds to the concept of weak solutions of PDEs, see [32,34,36]. In this article we aim to transfer yet another solution concept of PDEs to the case when the right-hand side of the PDE is perturbed by a stochastic noise term. This solution concept, see [26] for a comprehensive study and [28,37,42] for possible generalizations, is of operator-theoretic nature and takes place in an abstract Hilbert space setting. Its key features are establishing the time-derivative operator as a normal, continuously invertible operator on an appropriate Hilbert space and a positive definiteness constraint on the partial differential operator of the PDE (realized as an operator in space–time). Actually, this solution theory is a general recipe to solve a first-order (in time and space) system of coupled equations, and when solving a higher-order (S)PDE, it gets reduced to such a first-order system. In this sense the solution theory we will apply is roughly similar in spirit to the treatment of hyperbolic equations in [13,19], see also [2] for an application to SPDEs. We shall illustrate the class of SPDEs we will investigate using this approach. Throughout this article let H be a Hilbert space, that we think of as basis space for our investigation. We assume A to be a skew-self-adjoint, unbounded linear operator on H (i.e. i A is a self-adjoint operator on H ) which is thought of as containing the spatial derivatives. Furthermore, we denote by ∂0 the time-derivative operator that will be constructed as a normal and continuously invertible operator in Sect. 2.1. In particular, it can be shown that the spectrum of ∂0−1 is contained in a ball of the right half plane touching 0 ∈ C. Let for some r > 0, M : B(r, r ) → L(H ) be an analytic function, where B(r, r ) is the open ball in C with radius r > 0 centered at r > 0, and L(H ) the set of bounded linear operators on H . Then one can define via a functional calculus the linear operator M (∂0−1) as a function of the inverse operator ∂0−1, which will be specified below. The idea to define this operator is to use the Fourier–Laplace transform as explicit spectral representation as multiplication operator for ∂0 yielding a functional calculus for both ∂0 and its inverse. The role that M (∂0−1) plays is coupling the equations in the first-order system. In applications, M (∂0−1) also contains the information about the ‘constitutive relations’ or the ‘material law’. Throughout this article we consider the following (formal (...truncated)


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André Süß, Marcus Waurick. A solution theory for a general class of SPDEs, Stochastics and Partial Differential Equations: Analysis and Computations, 2017, pp. 278-318, Volume 5, Issue 2, DOI: 10.1007/s40072-016-0088-8