Stochastic differential equations and stochastic parallel translations in the Wasserstein space
HAL (Le Centre pour la Communication Scientifique Directe)(2023)
摘要
We will develop some elements in stochastic analysis in the Wasserstein space $\mathbb{P}_2(M)$ over a compact Riemannian manifold $M$, such as intrinsic It$\^o$ 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 $\mathbb{P}_2(\mathbb{T})$. 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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关键词
stochastic parallel translations,stochastic differential equations,wasserstein space
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