Abstract
In this paper, we study the set of stationary solutions of the Vlasov–Fokker–Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. We prove, under suitable assumptions, that the VFP equation does not have a unique stationary solution and that there exists a phase transition. Our study relies on the recent results by Tugaut and coauthors regarding the McKean–Vlasov equation.
Original language | English |
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Pages (from-to) | 38-45 |
Number of pages | 8 |
Journal | Applied Mathematics Letters |
Volume | 52 |
Early online date | 15 Aug 2015 |
DOIs | |
Publication status | Published - Feb 2016 |
Keywords
- Invariant measure
- Vlasov–Fokker–Planck equation
- McKean–Vlasov equation
- Stochastic processes