Some Characterizations And Estimation Of Harmonic Functions And Some Distributions For A Class Of Hybrid Systems

This paper mainly studies some characterizations and distribution estimations for a class of equations driven by stable processes with Markov switching.First of all,some sufficient conditions of maximum principle corresponding to the solutions of equation-s driven by stable processes satisfy have been provided.Furthermore,under certain conditions,the Maximum principle of weak solution for stable process with Markov switching is established by using the Green function method.As an application,some explicit distribution characterization and estimation problems of stochastic differential equations driven by Brownian motion with Markov switching are studied.By Bessel function,Laplace transform and time-changing methods,the explicit expression of its density function is obtained.The mean exit time and the first exit time of switching Brownian motion is obtained by solving the corresponding Poisson equation.By prod-uct,our results for the exit time reveal that when ?₁tends to ?₂,the Laplace transform of the mean exit time and the first exit time is affected.Finally,the appropriate up-per and lower bounds of the maximum estimation of Brownian motion with Markov switching are given.