Stochastic differential equations and stochastic parallel translations in the Wasserstein space

Abstract

We will develop some elements in stochastic analysis in the Wasserstein space P2(M)\mathbb{P}_2(M) over a compact Riemannian manifold MM, such as intrinsic Itoˆ\^o formulae, stochastic regular curves and parallel translations along them. We will establish the existence of parallel translations along regular curves, or stochastic regular curves in case of P2(T)\mathbb{P}_2(\mathbb{T}). Surprisingly enough, in this last case, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE

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