Pathwise uniqueness for stochastic heat and damped equations with Hölder continuous drift

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.