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 $W_F(\rho d x)=\int_{\mathbb R^d}F(x,\rho(x))d x$. 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 $\Lambda$ on $\mathcal P_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\geq 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $\Lambda$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $DW_F(\mu)$ is defined for $\Lambda$-a.e. $\mu$ and that the Gaussian-based reference measure $\Lambda$ can be chosen in such way that the distorted process ${(\mu_t)}_{t\geq 0}$ is a martingale solution for the equation $d\mu_t=-DW_F(\mu_t) d t+d R_t$, $t\geq 0$.