The stochastic-process interpretation conjecture for gradient-flow structures

The diffusion equation can be viewed as a gradient flow in several distinct ways, including as the $L^2$-gradient flow of the Dirichlet integral, as the $H^{-1}$-gradient flow of the $L^2$-norm, and as the $H^{s-1}$-gradient flow of the $H^s$-seminorm for each $s\in\mathbb R$. Each gradient-flow structure can be connected to an appropriate stochastic process via a large-deviation principle. This proposes a unifying stochastic interpretation of gradient-flow formulations: the underlying stochastic process should determine the associated dissipation and energy through its large deviations. The Wasserstein structure is motivated by Brownian particles, while the symmetric simple exclusion process motivates the mobility $\rho(1-\rho)$; the claim asserts that analogous connections exist for each gradient-flow structure.