Governing Equations Of Probability Density For Stochastic Differential Equations With Multiple Discrete Time Delays

Stochastic dynamical systems with time delay involve many research fields,such as biology,mechanical engineering,control systems,and so on.In the whole dynamic category,there are still many difficulties in the study of stochastic dynamical systems with discrete delays.At present,the research on stochastic dynamical system with time delay is mainly focused on the existence and stability of its analytic solution,but the theory of governing equation of probability density lacks relevant research.Therefore,it is of great significance to explore the governing equation of probability density for stochastic dynamical system with time delay.For stochastic differential equation without discrete delay,the probability density can be solved by Fokker Planck equation,which requires conjugate infinitesimal generator for the solutions.Because the stochastic differential equation with discrete delay is not only related to the current state,but also depends on the past state,it is not a Markov process.This kind of equation has no corresponding conjugate infinitesimal generator,so it is impossible to obtain its probability density equation by using Fokker Planck equation.Some scholars have given the governing equation of probability density for the stochastic differential equation with a single discrete delay to solve its density.In this paper,we mainly study the governing equation of probability density for stochastic differential equations with multiple discrete delays when the ratio of multiple discrete delays is rational,and generalize the previous conclusions to a certain extent.The method adopted is to express the governing equation of probability density for stochastic differential equations with multiple discrete delays by the governing equation of probability density without discrete delays.Because the governing equation of probability density for stochastic differential equations without discrete delays has been deeply studied and can be obtained by solving the corresponding Fokker-Planck equation.