scholarly journals The Linear Fokker-Planck Equation for the Ornstein-Uhlenbeck Process as an (Almost) Nonlinear Kinetic Equation for an Isolated N-Particle System

2006 ◽  
Vol 123 (3) ◽  
pp. 525-546 ◽  
Author(s):  
Michael Kiessling ◽  
Carlo Lancellotti
Keyword(s):  
Kinetic Equation ◽  
Particle System ◽  
Planck Equation ◽  
Uhlenbeck Process ◽  
Fokker Planck Equation ◽  
Ornstein Uhlenbeck Process ◽  
Nonlinear Kinetic ◽  
Fokker Planck

scholarly journals Time-changed fractional Ornstein-Uhlenbeck process

2020 ◽  
Vol 23 (2) ◽  
pp. 450-483 ◽  
Author(s):  
Giacomo Ascione ◽  
Yuliya Mishura ◽  
Enrica Pirozzi
Keyword(s):  
Planck Equation ◽  
Uhlenbeck Process ◽  
Fokker Planck Equation ◽  
Ornstein Uhlenbeck Process ◽  
Fokker Planck

AbstractWe define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process.

The Fokker–Planck equation for the time-changed fractional Ornstein–Uhlenbeck stochastic process

2021 ◽  
pp. 1-26
Author(s):  
Giacomo Ascione ◽  
Yuliya Mishura ◽  
Enrica Pirozzi
Keyword(s):  
Mild Solutions ◽  
Sufficient Conditions ◽  
Planck Equation ◽  
Strong Solutions ◽  
Uniqueness Result ◽  
Uhlenbeck Process ◽  
Weak Maximum Principle ◽  
Fokker Planck Equation ◽  
Ornstein Uhlenbeck Process ◽  
Fokker Planck

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

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

2013 ◽  
Vol 15 (05) ◽  
pp. 1350017 ◽  
Author(s):  
MARK A. PELETIER ◽  
D. R. MICHIEL RENGER ◽  
MARCO VENERONI
Keyword(s):  
Variational Formulation ◽  
Large Deviation ◽  
Particle System ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Stochastic Particle System ◽  
Deviation Rate ◽  
The Many ◽  
Particle Approach ◽  
Fokker Planck

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.

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

2019 ◽  
Vol 21 (04) ◽  
pp. 1850039 ◽  
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.

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

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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