Probability Density Evolution For Dynamical Systems Driven By Levy Process

The study of probability density evolution of stochastic dynamical systems is an important way to study the properties of stochastic dynamical systems and to simulate the related properties of stochastic systems.To study the probability density evolution of stochastic dynamical systems,we first study the dynamical systems driven by Levy process.On the one hand,people prefer to ignore the delay time ?,then the study of the stochastic dynamic system will become more simple.On the other hand,there have been some conclusion about the study of the probability density evolution for dynamical systems without time delays driven by Levy process.Therefore,the emphasis of this paper will be placed on the probability density evolution of a stochastic dynamic system with time delays.At first,we give some preparations about Levy process,stochastic differential equation and Fokker-Planck equation which will give a great effect in the study below.At second we give the corresponding assumptions and lemmas.We give a class of stochastic delay differential equations which have time delays both in drift coefficient and noise amplitude in order to characterize a class of stochastic dynamical systems with time delays.Finally,we use the perturbation expansion method and give the time-delay Fokker-Planck equation of the stochastic dynamical system driven by Brownian motion.We can not get the Fokker-Planck equation for stochastic delay differential equations through the general method,because the solutions to stochastic delay differential equations are not Markov processes and have no corresponding infinitesimal generators.We can construct a unique solution that corresponds to the solution of a stochastic differential equations without time delays by convening corresponding assumptions and constraints.Then,we can get its time-delay Fokker-Planck equation.Finally we get the time-delay Fokker-Planck of a stochastic delay differential equations driven by Levy process.In this paper,we get the time-delay Fokker-Planck of a stochastic delay differential equations driven by Levy process,which will give a great effect to study the stochastic time-delay system.Besides,it can provide convenience to study the stochastic system with more time delays.