# On Cherny's results in infinite dimensions: A theorem dual to Yamada-Watanabe

## TL;DR

This paper extends Cherny's finite-dimensional results to infinite-dimensional stochastic PDEs, establishing equivalences between various notions of uniqueness and existence of solutions in a Hilbert space framework.

## Contribution

It proves that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions of stochastic PDEs in infinite dimensions, generalizing Cherny's finite-dimensional findings.

## Key findings

- Joint uniqueness in law implies pathwise uniqueness.
- Existence of a strong solution implies joint uniqueness in law.
- In infinite dimensions, uniqueness in law is equivalent to joint uniqueness in law.

## Abstract

We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the form \begin{align*} \text{d}X_t=b(t,X)\text{d}t+\sigma(t,X)\text{d}W_t, \,\,\,t\geq 0, \end{align*} and show that for such equations uniqueness in law is equivalent to joint uniqueness in law. Here $W$ is a cylindrical Wiener process in a separable Hilbert space $U$ and the equation is considered in a Gelfand triple $V \subseteq H \subseteq E$, where $H$ is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of A. Cherny for the case of finite-dimensional equations.