Lipschitz estimates on the JKO scheme for the Fokker–Planck equation on bounded convex domains
Introduction
This short paper is concerned with the parabolic PDE 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 is supposed here to be given, independent of time and of , and sufficiently smooth. We consider this equation on a bounded domain , and we complete it with no-flux boundary conditions
Starting from the seminal paper [1], it is well-known that this equation is the gradient flow of the functional with respect to the Wasserstein distance . 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 and then iteratively solving defining 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 is close to .
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 to . The case where a same norm increases when passing from to , but no more than a quantity of the order of , so that the bound can be iterated, is also interesting. We cite for instance bounds (see for instance Section 7.4.1 in [3]), BV bounds (see [6]), which are valid when and the entropy is replaced by an arbitrary convex penalization .
We are interested in this paper in Lipschitz bounds and more precisely in the quantity . 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 if one sets ) if is convex. If one can see that decreases in time, which implies exponential convergence as if or controlled growth if .
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 which is proven to decrease along iterations if 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 and sending also provides an estimate on . The same computation unfortunately does not work if we only have with , since a similar estimate is only possible as soon as , which prevents to send without sending . In this sense, it is not an estimate on the JKO scheme. Moreover, obtaining an 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 to . 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 is not attained on the boundary when 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 be a bounded closed convex domain in , V a non-negative Lipschitz function on , and a probability density on (i.e. and ). We study the minimizer of the functional on (the set of probabilities measures on ) given by : where is the Wasserstein distance on .
We know that this functional admits a unique minimizer, denoted in the rest of the paper. Indeed, 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).
References (14)
On the Jordan-Kinderlehrer-Otto scheme
J. Math. Anal. Appl.
(2015)- et al.
The variational formulation of the Fokker–Planck equation
SIAM J. Math. Anal.
(1998) - et al.
Euclidean, metric, and Wasserstein gradient flows: an overview
Bull. Math. Sci.
(2017)- et al.
BV estimates in optimal transportation and applications
Arch. Ration. Mech. Anal.
(2016)