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)