A weak space-time formulation for the linear stochastic heat equation

Abstract: The topic covered in this thesis is the introduction of a new formulation for the linear stochastic heat equation driven by additive noise, based on the space-time variational formulation for its deterministic counterpart. Having a variational formulation allows the use of the so called inf-sup theory in order to obtain results of existence and uniqueness in a relatively simple way. The Banach-Necas-Babuska inf-sup theory is a well known functional analytic tool which has been lately used in connection to the linear heat equation. During the last years, indeed, there has been has been a renewed interest in this theory, that characterizes existence, uniqueness, and continuous dependence on data of the variational problem. The theory provides a variational formulation with different trial and test spaces ant it is therefore a natural basis for Petrov-Galerkin approximation methods. However, in most cases, its use has been limited only within a deterministic framework for the linear heat equation. In the appended paper we present a weak space-time formulation for the linear stochastic heat equation with additive noise, where both trial and test functions have a stochastic component and we give sufficient conditions on the the data and on the covariance operator associated to the noise in order to have existence and uniqueness of the solution. We show the connection between the obtained solution and the other concepts of solution present in literature and, with the further assumption that the elliptic operator in the stochastic heat equation does not have a stochastic component and is independent of time, we derive some properties regarding the spatial regularity. Finally, we present two possible semi-discrete schemes to compute an approximate solution, one in space and one in time, where, in particular, the time-stepping obtained using piecewise linear test functions and piecewise constant trial functions is a modification of the Crank-Nicolson scheme. In both cases we bound the errors by means of the quasi-optimality theory.