In this paper, a Vlasov-Fokker-Planck equation associated to a stochastic interacting particle system with distributed delay is analytically investigated. Under certain restrictions on the parameters, the well-posedness and equilibration of the Vlasov-Fokker-Planck equation are shown. Furthermore, an exponential rate of convergence toward an equilibrium measure is proven as long as the delay horizon is finite. For infinite delay horizon, i.e., when the entire history of the solution paths is taken into consideration, algebraic decay of the solution is shown.
- Delay equations
- Interacting stochastic particles
- Retarded mean-field equations
- Trend to equilibrium
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics