Pathwise uniqueness for stochastic heat and damped equations with H\"older continuous drift

TL;DR

This paper establishes pathwise uniqueness for certain infinite-dimensional stochastic PDEs, including heat, damped wave, and beam equations, without requiring the structure condition, extending previous results.

Contribution

It proves pathwise uniqueness for stochastic heat, damped wave, and beam equations in specific dimensions without the structure condition, broadening applicability.

Findings

Pathwise uniqueness proven for stochastic heat equation in dimension 3.

Pathwise uniqueness established for stochastic damped wave equation in dimension 1.

Results include stochastic Euler-Bernoulli damped beam equation up to dimension 3.

Abstract

In this paper, we prove pathwise uniqueness for stochastic differential equations in infinite dimension. Under our assumptions, we are able to consider the stochastic heat equation up to dimension $3$, the stochastic damped wave equation in dimension $1$ and the stochastic Euler-Bernoulli damped beam equation up to dimension $3$. We do not require that the so-called {\it structure condition} holds true.