On the Continuity of Pathwise Solutions to Langevin Equations in Infinite Dimensions

Abstract

We fix a rich probability space (Ω,F,P). Let (H,‖⋅‖) be a separable Hilbert space and let μ be the canonical cylindrical Gaussian measure μ on H. Given any abstract Wiener space (H,B,μ) over H, and for every Hilbert–Schmidt operator T: H⊂B→H which is (|{⋅}|,‖⋅‖)-continuous, where |{⋅}| stands for the (Gross-measurable) norm on B, we construct an Ornstein–Uhlenbeck process ξ: (Ω,F,P)×[0,1]→(B,|{⋅}|) as a pathwise solution of the following infinite-dimensional Langevin equation dξt=db t+T(ξt) dt with the initial data ξ0=0, where b is a B-valued Brownian motion based on the abstract Wiener space (H,B,μ). The richness of the probability space (Ω,F,P) then implies the following consequences: the probability space Ω is independent of the abstract Wiener space (H,B,μ) (in the sense that (Ω,F,P) does not depend on the choice of the Gross-measurable norm |{⋅}|) and the space C B consisting of all continuous B-valued functions on [0,1] is identical with the set of all paths of ξ. Finally, we present a way to obtain pathwise continuous solutions ξ: dξt= $$\sqrt {\left| \alpha \right|\beta }$$ db t+α⋅ξt dt with initial data ξ0=0, where α,β∈R,α≠0 and 0<β.