Pathwise uniqueness in infinite dimension under weak structure conditions

TL;DR

The paper addresses pathwise uniqueness for infinite-dimensional SDEs of the form $dX(t)=AX(t)dt+\widetilde{\mathscr L}B(X(t))dt+GdW(t)$ with Hölder-continuous drift and nonanalytic semigroups. Using a finite-dimensional approximation plus a refined Itô–Tanaka (regularization-by-noise) approach and a backward-Kolmogorov/Mild-solution framework, the authors establish Lipschitz dependence of weak mild solutions on the initial data, yielding pathwise uniqueness under weak structure conditions that involve an auxiliary operator $\Lambda$. They apply the abstract results to stochastic damped wave, damped Euler–Bernoulli beam, and heat equations, allowing damping-related regularity to compensate for rough drifts and without requiring Hilbert–Schmidt noise or trace-class conditions in many cases. The results extend known uniqueness paradigms to hyperbolic-type SPDEs and to lower-regularity noise, providing both existence (strong, via On03/On04-type arguments) and robust well-posedness that are pertinent for finite- and infinite-dimensional damped systems. Overall, the work broadens the class of infinite-dimensional SDEs with unique solutions and Lipschitz data-dependence, with direct implications for stochastic PDE models in physics and engineering.

Abstract

Let $U,H$ be two separable Hilbert spaces and $T>0$. We consider an SDE which evolves in the Hilbert space $H$ of the form \begin{align} dX(t)=AX(t)dt+\widetilde{\mathscr L}B(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 $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(Ω,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $B:H\to H$ is a bounded and $θ$-Hölder continuous function, for some suitable $θ\in(0,1)$, and $\widetilde{\mathscr L}:H\to H$ and $G:U\to H$ are linear bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to the equation depends on the initial datum in a Lipschitz way. This implies that pathwise uniqueness holds true. Here, the presence of the operator $Λ$ plays a crucial role. 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 upto dimension $3$ even in the hyperbolic case.

Theorems & Definitions (38)

• Definition 2.1
• Remark 3.2
• Remark 3.3
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Remark 3.7
• Theorem 3.8
• proof