Strong uniqueness for stochastic evolution equations with unbounded measurable drift term

TL;DR

This paper proves strong pathwise uniqueness for stochastic evolution equations in Hilbert spaces with unbounded, merely measurable, and locally bounded drift terms, extending previous results to more general conditions.

Contribution

It extends the strong uniqueness results for stochastic evolution equations to cases with unbounded, measurable, and locally bounded drifts, broadening the class of equations with guaranteed unique solutions.

Findings

Pathwise uniqueness holds for a large class of solutions.

Existence of strong solutions is established under minimal drift regularity.

The results generalize previous bounded drift assumptions to unbounded, measurable drifts.

Abstract

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $B$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner, Annals of Prob., published online in 2012) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in our previous paper pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift $B$ is only measurable, locally bounded and grows more than linearly.