Uniqueness for stochastic differential equations in Hilbert spaces with irregular drift

TL;DR

This paper introduces a new framework for proving strong existence and uniqueness of solutions to stochastic differential equations in Hilbert spaces with irregular, Hölder continuous drift, extending previous results without structural assumptions on the drift.

Contribution

It develops a novel approach combining stochastic sewing, Gaussian analysis, and approximation techniques to establish uniqueness for SDEs with irregular drift in infinite-dimensional spaces.

Findings

Proves strong uniqueness for SDEs with Hölder continuous drift in Hilbert spaces.

Extends the class of admissible drift regularity beyond previous results.

Introduces a new method avoiding infinite-dimensional Kolmogorov equations.

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)^{-\gamma/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 $\alpha$-H\"older continuous function $H\to H$, and a nonnegative parameter $\gamma$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $\alpha > \alpha^*(\gamma)$, with an explicit function $\alpha^*$ that takes values in $(0,1)$ for all $\gamma\in[0,3)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. The range of admissible $\alpha$ is also extended. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining L\^e's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.