Hidden dissipation and convexity for Kimura equations

TL;DR

This paper rigorously establishes a gradient flow framework for one-dimensional Kimura equations using Wasserstein-Shahshahani geometry, providing new insights into their long-term behavior and convergence properties.

Contribution

It completes the formal gradient flow formulation of Kimura equations by establishing a rigorous variational approach via conditioning and convexity analysis.

Findings

Derived contraction estimates for Kimura equations

Proved quantitative convergence to stationary distribution

Established a gradient flow structure for the equations

Abstract

In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic process to non-fixation in order to get rid of singularities on the boundaries, and then studying the conditioned $Q$-process from a more traditional and variational point of view. In doing so we complete the work initiated in [Chalub et Al., Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective. Acta App Math., 171(1), 1-50], where the gradient flow was identified only formally. The approach is based on the Energy Dissipation Inequality and Evolution Variational Inequality notions of metric gradient flows. Building up on some convexity of the driving entropy functional, we obtain new contraction estimates and quantitative long-time convergence towards the stationary distribution.