Pathwise uniqueness for a class of SDE in Hilbert spaces and applications

An abstract evolution equation in Hilbert spaces is considered. In the deterministic case, it includes several examples of non-uniqueness. It is proved that pathwise uniqueness is restored by means of a suitably non-degenerate additive noise. The proof is based on the associated infinite dimensional Kolmogorov equation. © 2009 Elsevier Inc. All rights reserved