On the Continuity of Pathwise Solutions to Langevin Equations in Infinite Dimensions |
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Authors: | Horst Osswald Jiang-Lun Wu |
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Affiliation: | (1) Mathematisches Institut der LMU-München, Theresienstr. 39, D-80333 München, Germany. e-mail;(2) Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom. e-mail |
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Abstract: | We fix a rich probability space (,F,P). Let (H,) be a separable Hilbert space and let be the canonical cylindrical Gaussian measure on H. Given any abstract Wiener space (H,B,) over H, and for every Hilbert–Schmidt operator T: HBH which is (|{}|,)-continuous, where |{}| stands for the (Gross-measurable) norm on B, we construct an Ornstein–Uhlenbeck process : (,F,P)×[0,1](B,|{}|) as a pathwise solution of the following infinite-dimensional Langevin equation dt=dbt+T(t)dt with the initial data 0=0, where b is a B-valued Brownian motion based on the abstract Wiener space (H,B,). The richness of the probability space (,F,P) then implies the following consequences: the probability space is independent of the abstract Wiener space (H,B,) (in the sense that (,F,P) does not depend on the choice of the Gross-measurable norm |{}|) and the space CB consisting of all continuous B-valued functions on [0,1] is identical with the set of all paths of . Finally, we present a way to obtain pathwise continuous solutions :dt=dbt+tdtwith initial data 0=0, where ,R,0 and 0<. |
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Keywords: | abstract Wiener spaces Langevin equations Ornstein– Uhlenbeck processes Hilbert– Schmidt operators continuity of pathwise solutions rich probability spaces Loeb measure spaces nonstandard analysis |
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