Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Abstract

In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional K h , which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, K h is asymptotically equivalent, in the sense of Γ -convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.1 E 1 is the Boltzmann-Shannon entropy. 0921-7134/15/$35.00 © 2015 -IOS Press and the authors. All rights reserved 86 M.H. Duong / Asymptotic equivalence of the discrete variational functional with respect to the Wasserstein distance W 2 on the space of probability measures with finite second moment P 2 (R d ),where Γ (μ, ν) is the set of all probability measures on R d × R d with marginals μ and ν. This statement can be understood in a variety of different ways. In [27], Otto shows this from a differential geometry point of view by considering (P 2 (R d ), W 2 ) as an infinite dimensional Riemannian manifold. However, for the purpose of this paper, the most useful way is that the solution t → ρ(t, x) can be approximated by the JKO-discretization scheme as follows [17,31].

A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

“…The ϕ-exponential measures play an important role in statistical physics, information geometry and in the analysis of nonlinear diffusion equations [26,25,32,33]. We refer to [25,32,13] for further details on q-Gaussian measures, ϕ-exponential measures and and their properties.…”

“…By Lemma 3.2 this equation has a positive definite solution. This together with the strict convexity of f imply that f has a unique minimizer which is a Gaussian measure N (0, X) where X solves (13). In the one dimensional case this equation reads…”

“…Suppose µ i are either Gaussian measures or q-Gaussian measures or ϕexponential measures (in this case, γ = 0) with mean zero. Then the minimization problem (1) has a unique minimizer whose variance solves the nonlinear matrix equation (13) or (17) or (27) respectively. Furthermore, the objective function is strictly convex.…”