Stochastic differential equations and stochastic parallel translations in the Wasserstein space

2023/09/15 by Ding, Hao, Shizan Fang, Fang, Shizan +2
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2309.08702

Abstract

We will develop some elements in stochastic analysis in the Wasserstein space ℙ₂(M) over a compact Riemannian manifold M, such as intrinsic Itô 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(\mathbbT). Surprisingly enough, in this last case, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE.

Related

• Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold
• An intrinsic parallel transport in Wasserstein space
• Stochastic geodesics
• The tangent space to the Wasserstein space: parallel transport and other applications
• Euler and Betti curves are stable under Wasserstein deformations of distributions of stochastic processes