Uniqueness for stochastic differential equations in Hilbert spaces with irregular drift

TL;DR

This work addresses strong well-posedness for stochastic evolution equations in a Hilbert space with irregular drift by combining stochastic sewing in Hilbert spaces, Gaussian averaging, and Lasry–Lions approximation. Under Hölder drift $b\in \mathcal{C}^\alpha(H,H)$ with exponent $\alpha\in(0,1]$, and a noise parameter $\gamma$ satisfying a trace condition and $\gamma<\frac{\alpha}{2-\alpha}$, the authors prove existence of a unique strong solution and a Lipschitz stability bound with respect to initial data; these results extend prior work without requiring structural drift assumptions. They apply the framework to the stochastic heat equation in dimensions $d\in\{1,2,3\}$ with Dirichlet-type boundary conditions, obtaining explicit admissible regimes for $\alpha$ and $\gamma$. The methodology avoids infinite-dimensional Kolmogorov equations and Zvonkin transforms, offering a versatile regularization-by-noise mechanism with potential extensions to time-dependent drifts and non-Markovian noise.

Abstract

We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A X_t + b(X_t))dt +(-A)^{-γ/2}dW_t,\quad X_0=x_0 \in H, \end{equation*} where $A$ is a self-adjoint negative definite operator with purely atomic spectrum, $W$ is a cylindrical Wiener process, $b$ is $α$-Hölder continuous function $H\to H$, and a nonnegative parameter $γ$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $α> 2γ/(1+γ)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining Lê's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.

Theorems & Definitions (18)

• Definition 2.2
• Theorem 2.3
• Corollary 2.4
• Remark 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof