VARIATIONAL FORMULATION OF THE FOKKER–PLANCK EQUATION WITH DECAY: A PARTICLE APPROACH

We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the many-particle limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Mosco-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker–Planck equation with decay.

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Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500396 ◽

2019 ◽

Vol 21 (04) ◽

pp. 1850039 ◽

Cited By ~ 2

Author(s):

José A. Carrillo ◽

Young-Pil Choi ◽

Samir Salem

Keyword(s):

Particle System ◽

Planck Equation ◽

Newtonian Potential ◽

Empirical Measure ◽

Propagation Of Chaos ◽

Interaction Forces ◽

Fokker Planck Equation ◽

Stochastic Particle System ◽

Stochastic Particle ◽

Fokker Planck

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

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Dynamical Systems in the Variational Formulation of the FokkerPlanck Equation by the Wasserstein Metric

Applied Mathematics & Optimization ◽

10.1007/s002450010008 ◽

2000 ◽

Vol 42 (2) ◽

pp. 203-227 ◽

Cited By ~ 4

Author(s):

T. Mikami

Keyword(s):

Dynamical Systems ◽

Variational Formulation ◽

Planck Equation ◽

Wasserstein Metric ◽

Fokker Planck Equation ◽

Fokker Planck

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Diffusion and Memory Effect in a Stochastic Processes and the Correspondence to an Information Propagation in a Social System

10.21203/rs.3.rs-792357/v1 ◽

2021 ◽

Author(s):

Peng Wang ◽

Jie Huo ◽

Xu-Ming Wang

Keyword(s):

Social System ◽

Particle System ◽

Planck Equation ◽

Generalized Langevin Equation ◽

Information Propagation ◽

Fokker Planck Equation ◽

Social Ideology ◽

Diffusion Particle ◽

With Memory ◽

Fokker Planck

Abstract A generalized Langevin equation is suggested to describe a diffusion particle system with memory. The equation can be transformed into the Fokker-Planck equation by using the Kramers-Moyal expansion. The solution of Fokker-Planck equation can describe not only the diffusion of particles but also that of opinion particles based on the similarities between the two. We find that the memory can restrain some non-equilibrium phenomena of velocity distribution in the system, without memory, induced by correlation between the noise and space[1]. However, the memory can enhance the effective collision among particles as shown by the variation of diffusion coefficients, and changes the diffusion mode between the dissipative and pumping region by comparing with that in the aforementioned system without memory. As the discussions in this physical system is paralleled to a social system, the random diffusion of social ideology, such as the information propagation, can be suppressed by the correlation between the noise and space.

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