Transport distances for PDEs: the coupling method

Abstract

We informally review a few PDEs for which some transport cost between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This means that we double the variables and solve, globally in time, a well-chosen PDE posed on the Euclidian square of the physical space. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.

Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

“…Finally, we refer to [9] for a simple proof of the equivalence between the contraction of the flow and the convexity condition, in which the gradient-flow structure of the problem is in fact not exploited. A related argument (coming from the probabilistic coupling method) can also be found in the recent manuscript [17].…”

A non-expanding transport distance for some structured equations

“…[ϕ(x, z, 0, r − εη) − ϕ(x, z, y, r)]k(dη) v s (dx, dz, dy, dr)ds. (23) Using ( 22) and ( 23) with a function ϕ depending only on (x, z) (or (y, r)), we see that the first marginal u 1 t (dx, dz) = (y,r)∈J v t (dx, dz, dy, dr) (and the second one u 2 t (dy, dr) = (x,z)∈J v t (dx, dz, dy, dr)) satisfies (20) and (21).…”

“…Our approach relies on the coupling method, see the survey paper [20]. The first example of use of the coupling method, to our knowledge, can be traced back to Dobrushin [14], where the nonlinear Vlasov equation is derived as mean-field limit of a deterministic system of interacting particles, making use of some transport cost.…”

Spaces of Measures and their Applications to Structured Population Models

“…For example, the asymptotics of transport-type equations can be linked to ergodic properties of Markov processes [27]. A connection between the two fields has been recently used to show nonexpansiveness of a semigroup of solutions in a transport distance, corresponding to the flat metric [21,22]. From a modelling perspective, a unified theory of well-posedness for models defined on a broad class of state spaces enables the analysis of state space structure through inverse problems using mechanistic models of biological processes and associated data.…”