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

We study a class of time-homogeneous diffusion processes on ℝn that share a common invariant measure but differ in their volatility matrices. In the Euclidean setting, we show that when the volatility matrix is the identity, the time-marginal distributions evolve as an entropic gradient flow in the quadratic Wasserstein space. This result recovers the gradient flow formulation of the Fokker–Planck equation, as established by Jordan, Kinderlehrer, and Otto. When ℝn is equipped with a Riemannian metric, we prove that the diffusion process becomes a gradient flow in the Wasserstein space induced by the metric. This characterization holds when the volatility matrix is the inverse of the metric tensor. Our approach combines stochastic control of the diffusion coefficient and time-reversal techniques. These findings align with results by Lisini, which build on the metric theory of Ambrosio, Gigli, and Savaré, and connect to Fathi’s work on large deviations for diffusion processes via gradient flows.