Weak uniqueness for stochastic partial differential equations in Hilbert spaces
Davide Addona, Davide Augusto Bignamini
TL;DR
This work proves weak uniqueness for a broad class of stochastic evolution equations in Hilbert spaces of the form $dX(t)=AX(t)dt+\mathcal{V}B(X(t))dt+GdW(t)$ under minimal regularity on $B$, by combining an Ornstein–Uhlenbeck regularizing framework with differentiability along the image of $\mathcal{V}$ and a finite-dimensional approximation strategy. The key ideas are to solve a stationary resolvent equation via a contraction mapping in the $\mathcal{V}$-differentiable setting, establish uniform finite-dimensional approximations, and pass to the infinite-dimensional limit to compare laws of two mild solutions. The main contributions are (i) weak well-posedness for large classes of heat and damped SPDEs in any dimension without Hölder continuity of $B$, (ii) a relaxed controllability-type condition on the noise that replaces previous assumptions, and (iii) concrete applicability to stochastic heat equations and damped wave/beam equations with explicit spectral/regularity requirements. This advances the probabilistic understanding of SPDEs in high dimensions by enabling weak uniqueness results under milder hypotheses than those required for pathwise uniqueness.
Abstract
Let $U,H$ be two separable Hilbert spaces. The main goal of this paper is to study the weak uniqueness of the Stochastic Differential Equation evolving in $H$ \begin{align*} dX(t)=AX(t)dt+\mathcal{V}B(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x \in H, \end{align*} where $\{W(t)\}_{t\geq 0}$ is a $U$-cylindrical Wiener process, $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup, $\mathcal{V},G:U\rightarrow H$ are linear bounded operators and $B:H\rightarrow U$ is a uniformly continuous function. The abstract result in this paper gives the weak uniqueness for large classes of heat and damped equations in any dimension without any Hölder continuity assumption on $B$.