  We prove pathwise uniqueness for a class of stochastic differential equations
(SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift
parts are sums of the sub-differential of a convex function and a bounded part.
This generalizes a classical result by one of the authors to infinite
dimensions. Our results also generalize and improve recent results by N.
Champagnat and P. E. Jabin, proved in finite dimensions, in the case where
their diffusion matrix is constant and nondegenerate and their weakly
differentiable drift is the (weak) gradient of a convex function. We also prove
weak existence, hence obtain unique strong solutions by the Yamada-Watanabe
theorem. The proofs are based in part on a recent maximal regularity result in
infinite dimensions, the theory of quasi-regular Dirichlet forms and an
infinite dimensional version of a Zvonkin-type transformation. As a main
application, we show pathwise uniqueness for stochastic reaction diffusion
equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a
unique strong solution.
