Consider the following SDE in a separable Hilbert space
$$ \mathrm dX_t = - A X_t \mathrm dt + f(t,X_t)\mathrm dt + \mathrm dW_t , $$
where $A$ is a positive, linear operator, $f$ is a bounded Borel measurable function and $W$ a cylindrical Wiener process. 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 SDE. This extends A. M. Davie’s result from $\mathbb R^d$ to Hilbert space-valued stochastic differential equations.
In this talk we consider the so-called path-by-path approach where the above SDE is considered as a random integral equation with parameter $\omega\in\Omega$. We show that there is a set $\Omega'$ of measure 1 such that for every $\omega\in\Omega'$ the corresponding integral equation for this $\omega$ has at most one solution. This notion of uniqueness (called path-by-path uniqueness) is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.