Diffusion processes as Wasserstein gradient flows via stochastic control of the volatility matrix

Abstract

We consider a class of time-homogeneous diffusion processes on $\mathbb{R}^{n}$ with common invariant measure but varying volatility matrices. In Euclidean space, we show via stochastic control of the diffusion coefficient that the corresponding flow of time-marginal distributions admits an entropic gradient flow formulation in the quadratic Wasserstein space if the volatility matrix of the diffusion is the identity. After equipping $\mathbb{R}^{n}$ with a Riemannian metric, we prove that the diffusion process can be viewed as a gradient flow in the inherited Wasserstein space if the volatility matrix is the inverse of the underlying metric tensor. In the Euclidean case, our probabilistic result corresponds to the gradient flow characterization of the Fokker$-$Planck equation, first discovered in a seminal paper by Jordan, Kinderlehrer, and Otto. In the Riemannian setting, the corresponding result on the level of partial differential equations was established by Lisini, building on the metric theory developed by Ambrosio, Gigli, and Savar\'{e}.