Lipschitz estimates on the JKO scheme for the Fokker–Planck equation on bounded convex domains

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Abstract

Given a semi-convex potential V on a convex and bounded domain Ω, we consider the Jordan–Kinderlehrer–Otto scheme for the Fokker–Planck equation with potential V, which defines, for fixed time step τ>0, a sequence of densities ρkP(Ω). Supposing that V is α-convex, i.e. D2VαI, we prove that the Lipschitz constant of logρ+V satisfies the following inequality: Lip(log(ρk+1)+V)(1+ατ)Lip(log(ρk)+V). This provides exponential decay if α>0, Lipschitz bounds on bounded intervals of time, which is coherent with the results on the continuous-time equation, and extends a previous analysis by Lee in the periodic case.

Introduction

This short paper is concerned with the parabolic PDE tρΔρ(ρV)=0,which is known as Fokker–Planck equation. It includes a linear diffusion effect (corresponding to the Laplacian in the above equation) and advection by a drift, which is supposed to be of gradient type. The function V is supposed here to be given, independent of time and of ρ, and sufficiently smooth. We consider this equation on a bounded domain Ω, and we complete it with no-flux boundary conditions (ρ+ρV)n=0.

Starting from the seminal paper [1], it is well-known that this equation is the gradient flow of the functional ρE(ρ)ρlogρ+Vdρwith respect to the Wasserstein distance W2. This distance is the one defined from the minimal cost in the optimal transport with quadratic cost. The reader can refer to [2], [3] for more details about optimal transport and Wasserstein distances, and [4], [5] for gradient flows in this setting.

The same paper [1] also provides a time-discretized approach to the above equation, now known as the Jordan–Kinderlehrer–Otto scheme (JKO). It consists in fixing a time step τ>0 and then iteratively solving minW22(ρ,ρk)2τ+E(ρ):ρP(Ω),defining ρk+1 as the unique minimizer of the above optimization problem. The sequence obtained in this way represents a good approximation of the continuous-time solution, in the sense that ρk is close to ρ(kτ,).

For this equation, which is linear and quite simple, using the JKO scheme as a tool to approximate the solution (for numerical or theoretical purposes) is not necessary, but the same scheme can be useful for equations which are gradient flow of more involved functionals. In order to prove convergence, it is often necessary to prove compactness estimates on the solutions of the scheme. Among the bounds that one can prove, the most striking ones are those where a certain norm decreases when passing from ρk to ρk+1. The case where a same norm increases when passing from ρk to ρk+1, but no more than a quantity of the order of τ, so that the bound can be iterated, is also interesting. We cite for instance L bounds (see for instance Section 7.4.1 in [3]), BV bounds (see [6]), which are valid when V=0 and the entropy ρlogρ is replaced by an arbitrary convex penalization f(ρ).

We are interested in this paper in Lipschitz bounds and more precisely in the quantity Lip(logρ+V). Using for instance the well-known Bakry–Emery theory (see [7]) this quantity decreases in time along solutions of the continuous equation (which is a re-writing of the transport-diffusion equation tu=ΔuVu if one sets u=ρeV) if V is convex. If D2VαI one can see that eαtLip(logρt+V) decreases in time, which implies exponential convergence as t if α>0 or controlled growth if α<0.

It is highly remarkable that these bounds can exactly be translated into the JKO setting, i.e. they also holds along the discrete-time iterations of the scheme. This was already observed in [8], and the analysis is very similar to ours, but was only restricted to the periodic setting. The main object of the present paper is to extend the analysis of [8] to a case with boundary. This recalls what is done in [9], where a similar estimate is performed on the quantity |ρρ+V|pdρ,which is proven to decrease along iterations if V is convex and the domain is also convex (this was obtained thanks to the so-called five gradients inequality introduced in [6]). Taking the power 1p and sending p also provides an estimate on ρρ+VL. The same computation unfortunately does not work if we only have D2VαI with α<0, since a similar estimate is only possible as soon as 1+ατp>0, which prevents to send p without sending τ0. In this sense, it is not an estimate on the JKO scheme. Moreover, obtaining an L bound (i.e. an estimate on a maximal value) out of a limit of integral quantities and using the five gradients inequality is a complicated procedure which can be highly simplified, which is the scope of this note.

As in [8], we will obtain the desired estimate by taking the maximal point of the squared norm of a gradient and using the optimality conditions together with the Monge–Ampère equation on the optimal transport map from ρk+1 to ρk. Even if we believe that the presentation that we give here is lighter and if the regularity assumptions on the date have been reduced to the minimal ones, the computations that we present are exactly the same as those of [8] in the case of a torus. The added value of our work is the attention to the boundary. The key points are

  • a lemma which guarantees that max|φ| is not attained on the boundary when φ is the Kantorovich potential between two smooth densities on a ball;

  • an extension-approximation procedure to pass from an arbitrary convex domain to a larger ball, and to regularize the densities.

Section snippets

Main result

Let Ω be a bounded closed convex domain in Rd, V a non-negative Lipschitz function on Ω, τ>0 and g a probability density on Ω (i.e. gL1(Ω),g0 and Ωg=1). We study the minimizer of the functional Fg,Vτ on P(Ω) (the set of probabilities measures on Ω) given by : Fg,Vτ(ρ)=W22(ρ,g)2τ+E(ρ)=W22(ρ,g)2τ+Ωρlogρ+ΩVdρwhere W2 is the Wasserstein distance on P(Ω).

We know that this functional admits a unique minimizer, denoted ρ in the rest of the paper. Indeed, F is the sum of three convex functionals,

The proof

Proof

We follow the same idea as in [8] (Lemma 3.2), except that we need to get rid of a possible maximum on the boundary. We eliminate this case when the domain is a ball and we do the general case by approximation.

Acknowledgment

The second author acknowledges the support of the Monge-Ampère et Géométrie Algorithmique project, funded by Agence nationale de la recherche (ANR-16-CE40-0014 - MAGA).

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