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

Consider the stochastic differential equation $\mathrm dX_t = -A X_t \,\mathrm dt + f(t, X_t) \,\mathrm dt + \mathrm dB_t$ 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.