Wasserstein Diffusion on Multidimensional Spaces

2024/01/23 by Karl‐Theodor Sturm, Sturm, Karl-Theodor
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2401.12721

Abstract

Given any closed Riemannian manifold M, we construct a reversible diffusion process on the space \mathcal P(M) of probability measures on M that is (i) reversible w.r.t.~the entropic measure \mathbb P^β on \mathcal P(M), heuristically given as dℙ^β(μ)=(1)/(Z) e^{-β Ent(μ| m)} dℙ^*(μ); (ii) associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus \mathcal E_W(f)=\liminf_{g→ f} \frac12∫_\mathcal P(M) ‖∇_W g‖²(μ) d\mathbb P^β(μ); (iii) non-degenerate, at least in the case of the n-sphere and the n-torus.

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