# JKO schemes with general transport costs

## TL;DR

This paper modifies a mathematical scheme to use general transport costs on manifolds, enabling convergence to a Riemannian Fokker–Planck equation.

## Contribution

The paper introduces a modified JKO scheme using general transport costs that induce a Riemannian metric.

## Key findings

- The modified JKO scheme converges to the Riemannian Fokker–Planck equation under suitable conditions.
- The approach allows using simpler cost functions when the Riemannian distance is computationally difficult.
- The method is applied to Hessian manifolds using Bregman divergence as a cost function.

## Abstract

We modify the JKO scheme, which is a time discretization of the Wasserstein gradient flow, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges, under suitable conditions, to the corresponding Riemannian Fokker–Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker–Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence.