scholarly journals Large time behaviors of upwind schemes and $B$-schemes for Fokker-Planck equations on $\mathbb {R}$ by jump processes

2020 ◽  
Vol 89 (325) ◽  
pp. 2283-2320 ◽  
Author(s):  
Lei Li ◽  
Jian-Guo Liu
Keyword(s):  
Large Time ◽  
Jump Processes ◽  
Upwind Schemes ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Solvability of Kolmogorov-Fokker-Planck equations for vector jump processes and occupation time on hypersurfaces

2001 ◽  
Vol 28 (11) ◽  
pp. 637-652
Author(s):  
N. G. Dokuchaev
Keyword(s):  
Wide Class ◽  
Generalized Functions ◽  
Piecewise Smooth ◽  
Occupation Time ◽  
Jump Processes ◽  
Fokker Planck Equations ◽  
Tanaka Formula ◽  
Fokker Planck

We study occupation time on hypersurface for Markovn-dimensional jump processes. Solvability and uniqueness of integro-differential Kolmogorov-Fokker-Planck with generalized functions in coefficients are investigated. Then these results are used to show that the occupation time on hypersurfaces does exist for the jump processes as a limit in variance for a wide class of piecewise smooth hypersurfaces, including some fractal type and moving surfaces. An analog of the Meyer-Tanaka formula is presented.

scholarly journals Fokker–Planck representations of non-Markov Langevin equations: application to delayed systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180131 ◽  
Author(s):  
Luca Giuggioli ◽  
Zohar Neu
Keyword(s):  
Langevin Equation ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Delay Systems ◽  
Langevin Equations ◽  
Delayed Systems ◽  
Temporal Features ◽  
Fokker Planck Equations ◽  
Random Fluctuations ◽  
Fokker Planck

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker–Planck equations for the n -time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n  = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker–Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

Modified Fokker-Planck equations

1978 ◽  
Vol 36 (1) ◽  
pp. 65-78 ◽  
Author(s):  
Glenn T. Evans
Keyword(s):  
Fokker Planck Equations ◽  
Fokker Planck
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