Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 3277-3298 |
| Number of pages | 22 |
| Journal | Siam Journal on Mathematical Analysis |
| Volume | 49 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 29 Aug 2017 |
Bibliographical note
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 ([email protected]). ‡Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom ([email protected]). §Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600MB Eindhoven, The Netherlands ([email protected]).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
Funding
∗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 ([email protected]). ‡Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom ([email protected]). §Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600MB Eindhoven, The Netherlands ([email protected]).
Keywords
- Delay equations
- Fibers
- Interacting stochastic particles
- Retarded mean-field equations
- Trend to equilibrium
- Vlasov-Fokker-Planck
ASJC Scopus subject areas
- Analysis
- Computational Mathematics
- Applied Mathematics