Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization

TL;DR

This paper proves exponential convergence of a convexifying evolution equation and shows that a non-autonomous gradient flow's trajectories converge to the convex envelope's minimizers, advancing understanding of convexification dynamics.

Contribution

It introduces a stochastic control representation for the convexifying equation and establishes exponential convergence in the Lipschitz norm, along with convergence results for a non-autonomous gradient flow.

Findings

Solution converges exponentially in time to the convex envelope.

Non-autonomous gradient flow trajectories converge to convex envelope minimizers.

Stochastic control representation provides new insights into convexification dynamics.

Abstract

We consider an evolution equation similar to that introduced by Vese and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope.