Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term

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 et al. in Ann Probab 41:3306–3344, 2013) which generalized Veretennikov’s fundamental result to infinite dimensions assuming boundedness of the drift term. As in Da Prato et al. (Ann Probab 41:3306–3344, 2013), 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 assumed only to be measurable and bounded and grow more than linearly.