Weak convergence for the stochastic heat equation driven by Gaussian white noise
Xavier Bardina, Universitat Autònoma de Barcelona
Maria Jolis, Universitat Autònoma de Barcelona
Lluís Quer-Sardanyons, Universitat Autònoma de Barcelona
Abstract
In this paper, we consider a quasi-linear stochastic heat
equation with spatial dimension one, with Dirichlet boundary conditions
and controlled by the space-time white noise. We formally replace the
random perturbation by a family of noisy inputs depending on a
parameter that approximate the white noise in some sense. Then, we
provide sufficient conditions ensuring that the real-valued
mild solution of the SPDE perturbed by this family of noises
converges in law, in the space of continuous functions, to the
solution of the white noise driven SPDE. Making use of a suitable
continuous functional of the stochastic convolution term, we show that
it suffices to tackle the linear problem. For this, we prove that the
corresponding family of laws is tight and we identify the limit law by
showing the convergence of the finite dimensional distributions. We
have also considered two particular families of noises to that our
result applies. The first one involves a Poisson process in the plane
and has been motivated by a one-dimensional result of Stroock. The
second one is constructed in terms of the kernels associated to the
extension of Donsker's theorem to the plane.
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