Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold

2025/12/02 by Martin, Aymeric
#22E65 (Secondary) #49Q22 #53C30 #60D05 (Primary) #60G44 #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.02975

Abstract

In this work, we establish the existence of solutions to stochastic differential equations on the Wasserstein space over a closed Riemannian manifold, under suitable regularity assumptions on the driving vector fields. Interpreting the diffeomorphism group \mathscrD as a Riemannian submersion onto the smooth Wasserstein space ¶_∞, we further prove the existence and uniqueness of the stochastic parallel parallel transport along diffusions on ¶_∞. Finally, we show that equivariant diffusions on \mathscrD endowed with a principal bundle structure over ¶_∞ admit a unique factorization into a horizontal diffusion and a vertical component expressed as a right exponential of a process taking values in the Lie algebra \mathfrakg of the group G of volume preserving diffeomorphisms.

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