Asymptotic behaviors of stochastic periodic differential equation with Markovian switching

摘要

Some asymptotic behaviors of hybrid stochastic periodic differential equations are investigated in this paper. Firstly, by constructing the space of the periodic functions with probability measure values, we prove that the solution processes converge to a stochastic process with periodic distribution under the conditions of the existence and uniqueness, tightness of the transition probability and global attractivity of the solutions for stochastic periodic differential equations with Markovian switching. Then we give the definition of stochastic periodic solution of stochastic periodic differential equations, which can be regarded as the stochastic counterpart of the periodic solution for the deterministic systems. For the stochastic periodic logistic equation, we firstly give the expression of the unique explicit solution and other properties such as p-moment boundedness, global attractivity and asymptotic stability in distribution, we then prove the existence, uniqueness, and stability of the unique stochastic periodic solution. Finally, we simulate the sample trajectory of stochastic periodic logistic equation. The results show a certain degree of periodicity, in fact it is a simulation result for the mixture of periodicity and randomness.