Trend to equilibrium for a delay Vlasov-Fokker-Planck equation and explicit decay estimates

TL;DR

This paper analyzes a delay Vlasov-Fokker-Planck equation linked to a stochastic particle system, establishing conditions for well-posedness, ergodicity, and explicit decay rates towards equilibrium depending on the delay length.

Contribution

It provides the first analytical proof of exponential and polynomial decay rates for the delay Vlasov-Fokker-Planck equation under finite and infinite delay conditions.

Findings

Exponential convergence to equilibrium for finite delay

Polynomial decay for infinite delay

Conditions ensuring well-posedness and ergodicity

Abstract

In this paper, a delay Vlasov-Fokker-Planck equation associated to a stochastic interacting particle system with delay is investigated analytically. Under certain restrictions on the parameters well-posedness and ergodicity of the mean-field equation are shown and an exponential rate of convergence towards the unique stationary solution is proven as long as the delay is finite. For infinte delay i.e., when all the history of the solution paths are taken into consideration polynomial decay of the solution is shown.