TY - JOUR

T1 - Trend to equilibrium for a delay Vlasov-Fokker-Planck equation and explicit decay estimates

AU - Klar, Axel

AU - Kreusser, Lisa

AU - Tse, Oliver

N1 - Funding Information:
∗Received by the editors December 22, 2015; accepted for publication (in revised form) June 28, 2017; published electronically August 29, 2017. http://www.siam.org/journals/sima/49-4/M105402.html Funding: The work of the authors was supported by the Deutsche Forschungsgemeinschaft (DFG) within the RTG GrK 1932 “Stochastic Models for Innovations in the Engineering Sciences.” †Department of Mathematics, Kaiserslautern University of Technology, 67663 Kaiserslautern, Germany (klar@mathematik.uni-kl.de). ‡Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom (l.m.kreusser@damtp.cam.ac.uk). §Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600MB Eindhoven, The Netherlands (o.t.c.tse@tue.nl).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/8/29

Y1 - 2017/8/29

N2 - 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.

AB - 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.

KW - Delay equations

KW - Fibers

KW - Interacting stochastic particles

KW - Retarded mean-field equations

KW - Trend to equilibrium

KW - Vlasov-Fokker-Planck

UR - http://www.scopus.com/inward/record.url?scp=85028589942&partnerID=8YFLogxK

U2 - 10.1137/15M105402X

DO - 10.1137/15M105402X

M3 - Article

AN - SCOPUS:85028589942

VL - 49

SP - 3277

EP - 3298

JO - Siam Journal on Mathematical Analysis

JF - Siam Journal on Mathematical Analysis

SN - 0036-1410

IS - 4

ER -