Trend to Equilibrium for a Delay Vlasov--Fokker--Planck Equation and Explicit Decay Estimates

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.

Convergence to Equilibrium in Wasserstein Distance for Damped Euler Equations with Interaction Forces

“…There, the authors used entropy dissipation methods to heuristically derive functional inequalities that provided the decay rates to equilibrium under relatively strong global regularity assumptions on (̺, u). The results in [48] indicate a convergence behavior similar to spatially inhomogeneous entropy-dissipating kinetic equations where hypocoercivity of the operators involved played an important role in determining convergence to equilibrium [30,32,33,47,46,61]. There, the exponential decay rate λ = λ(γ) has the property that λ → 0 as γ → 0 and γ → +∞, i.e., the best equilibration rate for (1) holds for some γ ∈ (0, ∞).…”