scholarly journals Properties of stochastic integro-differential equations with infinite delay: Regularity, ergodicity, weak sense Fokker–Planck equations

2016 ◽  
Vol 126 (10) ◽  
pp. 3102-3123 ◽  
Author(s):  
Hongwei Mei ◽  
George Yin ◽  
Fuke Wu
Keyword(s):  
Differential Equations ◽  
Infinite Delay ◽  
Weak Sense ◽  
Fokker Planck Equations ◽  
Fokker Planck

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

scholarly journals Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shao-Hong Yan ◽  
Xiao-Hong Chen ◽  
Gong-Nan Xie ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Present Method ◽  
Decomposition Method ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Fokker Planck Equations ◽  
Fokker Planck

The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

scholarly journals Fractional Fokker-Planck-Kolmogorov type equations and their associated stochastic differential equations

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Marjorie Hahn ◽  
Sabir Umarov
Keyword(s):  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Order ◽  
Kolmogorov Equations ◽  
Kolmogorov Type ◽  
Fokker Planck Equations ◽  
Fokker Planck

AbstractThere is a well-known relationship between the Itô stochastic differential equations (SDEs) and the associated partial differential equations called Fokker-Planck equations, also called Kolmogorov equations. The Brownian motion plays the role of the basic driving process for SDEs. This paper provides fractional generalizations of the triple relationship between the driving process, corresponding SDEs and deterministic fractional order Fokker-Planck-Kolmogorov type equations.

Equivalent Stochastic Systems

1988 ◽  
Vol 55 (4) ◽  
pp. 918-922 ◽  
Author(s):  
Y. K. Lin ◽  
G. Q. Cai
Keyword(s):  
Probability Distribution ◽  
Stochastic Systems ◽  
Initial Conditions ◽  
Weak Sense ◽  
Wide Sense ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Probability Densities ◽  
Fokker Planck Equations ◽  
Fokker Planck

Equivalent stochastic systems are defined as randomly excited dynamical systems whose response vectors in the state space share the same probability distribution. In this paper, the random excitations are restricted to Gaussian white noises; thus, the system responses are Markov vectors, and their probability densities are governed by the associated Fokker-Planck equations. When the associated Fokker-Planck equations are identical, the equivalent stochastic systems must share both the stationary probability distribution and the transient nonstationary probability distribution under identical initial conditions. Such systems are said to be stochastically equivalent in the strict (or strong) sense. A wider class, referred to as the class of equivalent stochastic systems in the wide (or weak) sense, also includes those sharing only the stationary probability distribution but having different Fokker-Planck equations. Given a stochastic system with a known probability distribution, procedures are developed to identify and construct equivalent stochastic systems, both in the strict and in the wide sense.

Moment and Fokker-Planck equations for the growth and decay of small objects

1980 ◽  
Vol 371 (1747) ◽  
pp. 553-567 ◽  
Keyword(s):  
Differential Equations ◽  
Equilibrium Model ◽  
Continuity Equation ◽  
Rate Equations ◽  
Approach To Equilibrium ◽  
Discrete Equations ◽  
Analytical Results ◽  
Static Solutions ◽  
Fokker Planck Equations ◽  
Fokker Planck

Some aspects of the growth and decay of distributions of small objects, such as particles, bubbles, or droplets, that are governed by discrete rate equations, are investigated. Differential equations for the moments of the distribution are derived and are solved in an equilibrium model both for the static moments and for the approach to equilibrium. Comparisons with numerical and exact analytical results show that the method gives fairly accurate values for the moments. The method also indicates types of growth and decay problems where the replacement of the discrete equations by the continuity equation is justified. The accuracy of static solutions of various Fokker-Planck equations is also examined in the equilibrium model. It is found that a form proposed by Goodrich gives by far the best representation of the solution of the discrete equations.

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