Gradient flow structure for some nonlocal diffusion equations

Abstract

We study ``nonlocal diffusion equations'' of the form \[ \partial_{t}\frac{d\rho_{t}}{d\pi}(x)+\int_{X}\left(\frac{d\rho_{t}}{d\pi}(x)-\frac{d\rho_{t}}{d\pi}(y)\right)\eta(x,y)d\pi(y)=0\qquad(\dagger) \] where $X$ is either $\mathbb{R}^{d}$ or $\mathbb{T}^{d}$, $\pi$ is a probability distribution on $X$, and $\eta(x,y)$ is a ``transition kernel'' which may be singular as $x\rightarrow y$. For a suitable notion of weak solutions which we discuss below, we show that solutions to these nonlocal diffusion equations can be interpreted as gradient flows of the relative entropy with respect to a certain nonlocal Wasserstein-type metric defined in terms of $\eta$ and $\pi$. These ``nonlocal Wasserstein metrics'' endow the space of probability measures on $X$ with a formal Riemannian structure, thereby providing for us a nonlocal analogue of the \emph{Otto calculus} originally developed in the context of the 2-Wasserstein metric. The class of equations $(\dagger)$ includes a family of ``nonlocal Fokker-Planck equations'', which are thus identified as nonlocal Wasserstein gradient flows of the relative entropy, analogously with the usual Fokker-Planck equation and the $W_{2}$ metric. The gradient flow structure we provide allows us to deduce: existence and uniqueness of solutions to ($\dagger$) in a suitable class of weak solutions; stability of solutions in the sense of evolutionary $\Gamma$-convergence, with respect to perturbations of initial condition, reference measure $\pi$, and transition kernel $\eta$; sufficient conditions for exponential convergence to equilibrium, in terms of a nonlocal analogue of the log-Sobolev inequality; as well as the consistency of a finite-volume-type spatial discretization scheme in the $\mathbb{T}^{d}$ case.