Weak uniqueness for stochastic partial differential equations in Hilbert spaces

TL;DR

This paper establishes weak uniqueness for a class of stochastic partial differential equations in Hilbert spaces, including heat and damped equations, without requiring H"older continuity of the nonlinear term.

Contribution

It proves weak uniqueness for SPDEs in Hilbert spaces with broad classes of equations, removing the need for H"older continuity on the nonlinear component.

Findings

Weak uniqueness holds for heat and damped equations in any dimension.

No H"older continuity assumption on the nonlinear function B is needed.

Results apply to large classes of SPDEs in Hilbert spaces.

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 H\"older continuity assumption on $B$.