Weak convergence of stochastic integrals with applications to SPDEs
Xavier Bardina, Salim Boukfal
TL;DR
The work addresses the weak convergence of parameter-dependent stochastic integrals driven by kernels θ_n toward Brownian-sheet-driven integrals in a multi-parameter setting. It introduces a main theorem establishing sufficient conditions—on the integrand f, the moment bounds of θ_n, and the convergence of the primitive processes—to ensure convergence of X_n(x)=∫_D f(x,y)θ_n(y) dy to X(x)=∫_D f(x,y)W(dy) in the space C(D). The approach combines Kolmogorov-type tightness with convergence of finite-dimensional distributions via Cramer-Wold and Wiener integral isometry, including convergence on simple functions. As an application, it proves weak convergence of solutions to the stochastic Poisson equation with Dirichlet boundary conditions by approximating the noise with θ_n and employing a continuous fixed-point map for the mild solution, extending previous results from single-parameter SPDEs to the Poisson setting.
Abstract
In this paper we provide sufficient conditions for sequences of random fields of the form $\int_{D} f(x,y) θ_n(y) dy$ to weakly converge, in the space of continuous functions over $D$, to integrals with respect to the Brownian sheet, $\int_{D} f(x,y)W(dy)$, where $D \subset \mathbb{R}^d$ is a rectangular domain, $x \in D$, $f$ is a function satisfying some integrability conditions and $\{θ_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,x]}θ_n(y)dy$ converge in law to the Brownian sheet (in the sense of the finite dimensional distribution convergence). We then apply these results to stablish the weak convergence of solutions of the stochastic Poisson equation.