Path-by-path uniqueness of infinite-dimensional stochastic differential equations

Abstract

Consider the stochastic differential equation $\mathrm{d}{X}_t=-A{X}_t\mathrm{d}t+f(t,X_t)\mathrm{d}t+\mathrm{d}{B}_t$ in a (possibly infinite-dimensional) separable Hilbert space, where $B$ is a cylindrical Brownian motion and $f$ is a just measurable, bounded function. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this stochastic differential equation. This extends A. M. Davie's result from ${\mathbb{R}}^d$ to Hilbert space-valued stochastic differential equations.