Path-by-Path Uniqueness in Infinite Dimensions

Let $f$ be a bounded measurable function and $W$ a Wiener process. The question whether the integral equation $$X_t(\omega )=x_0+\underset{0}{\overset{t}{\int }}f(X_s(\omega ))\mathrm{d}s+W_t(\omega )$$ has at most one solution $X_t$ for almost all $\omega$ (so-called path-by-path uniqueness) has been posed by N.~Krylov and answered affirmatively by A.~Davie in ${\mathbb{R}}^d$.

This result can be understood as a ``regularisation by noise'' effect since in the absence of noise the above result fails to hold. Let $A$ be a positive linear operator on a separable Hilbert and $W$ a cylindrical Wiener process.

In this talk we consider the equation $$\mathrm{d}{X}_t=-A{X}_t\mathrm{d}t+f(t,X_t)\mathrm{d}t+\mathrm{d}{W}_t,$$ where $f$ 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 giving a positive answer to N.~Krylov's original question for SDEs taking values in an abstract Hilbert spaces for small nonlinearties $f$.

Naturally, this notion of uniqueness is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.