Stochastic damped wave and Euler-Bernoulli equations with singular locally Hölder continuous drift

Abstract

Let $U$ and $H$ be two separable Hilbert spaces and $T>0$. We consider a stochastic differential equation which evolves in the Hilbert space $H$ of the form \begin{align} \label{SDEa} dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA}{)}_{t\ge 0}$, $W=(W(t){)}_{t\ge 0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(\Omega ,\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\in [0,T]},\mathbb{P})$, $G:U\to H$ is a linear bounded operator and $B:[0,T]\times H\to H_{\beta }$ is a locally $\theta$-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable $\theta \in (0,1)$. Here, $H_{\beta }$ is a Hilbert space which contains $H$ with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space $H$ into itself. The coefficient $\beta \ge 0$ measures the order of such a singularity and the case $\beta =0$ corresponds to a drift term with values in $H$. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. In particular, the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler--Bernoulli beam equation up to dimension $3$, even in the hyperbolic case.