Stochastic intrinsic gradient flows on the Wasserstein space

Abstract

We construct stochastic gradient flows on the $2$-Wasserstein space ${\mathcal{P}}_2$ over ${\mathbb{R}}^d$ for energy functionals of the type $W_F(\ddot{I}dx)={\int }_{{\mathbb{R}}^d}F(x,\ddot{I}(x))dx$. The functions $F$ and ${\partial }_{2}F$ are assumed to be locally Lipschitz on ${\mathbb{R}}^d\times (0,\infty )$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First we define a class of Gaussian-based measures $\hat{I}›$ on ${\mathcal{P}}_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\ge 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $\hat{I}›$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $D{W}_F(\hat{I}¼)$ is defined for $\hat{I}›$-a.e. $\hat{I}¼$ and that the Gaussian-based reference measure $\hat{I}›$ can be chosen in such way that the distorted process ${(\hat{I}{¼}_t)}_{t\ge 0}$ is a martingale solution for the equation $d\hat{I}{¼}_t=-D{W}_F(\hat{I}{¼}_t)dt+d{R}_t$, $t\ge 0$.