Periodic Measures Of Random Periodic Processes And Stochastic Differential Equations

Random processes can be used to model many natural and social systems,especially Markov processes.Many systems in the real world are perturbed by discontinuous noises,which can be modelled by stochastic differential equations driven by Levy processes with jumps.On the other hand,many systems in the world have some kind of repeating pattern.Randomness and periodicity coexist and interwine to form random periodicity.The periodic measure is the counterpart of the time period of the invariant measure,which can be used to describe the long-term asymptotic periodic behavior of stochastic systems.Based on this,in this thesis we study periodic measures of random periodic processes and periodic stochastic differential equations driven by α-stable processes.For random periodic processes,we mainly study periodic Markov chains and periodic Markov processes.Firstly,we prove that there is a unique periodic measure and an invariant measure for periodic irreducible time-homogeneous Markov chains on a finite state spaces,and give the relationship between the periodic measures and the invariant measures,and show that the average of the periodic measure over a period is an invariant measure.Secondly,we obtain the same conclusion for the periodic irreducible time-homogeneous Markov chains by lifting the periodic irreducible time-inhomogeneous Markov chains to a random process on a cylinder.Then,we use the minimum non-negative solution theory to give the simeient condition for the existence and uniqueness of periodic measures for periodic irreducible time-homogeneous Markov chains on a finite state space.Finally,we prove that periodic irreducible strong Feller’s Markov process on a locally compactly separable metric space has a unique periodic measure under the f-norm,and the periodic measure is geometrically ergodic under the f-norm based on the Lyapunov function.For periodic stochastic differential equations,we mainly study time-periodic stochastic differential equations with singular drift coefficients driven by symmetric rotationally invariant α-stable processes.Under some certain assumptions,we firstly prove that there is a unique regular strong solution to the stochastic differential equations.Secondly,we use the existence and uniqueness of the solution to prove that the Markov transition function P(s,t,x,·)defined by the solution of the equation is periodic with respect to the time variable.Then,we prove that the Markov semigroup P(s,t)is irreducible and strong Feller based on its heat kernel eatimation.Finally,we give the existence,uniqueness and geometric ergodicity of periodic measures of periodic stochastic differential equations under the f-norm.In particular,we apply the results to periodic stochastic differential equations satisfying a certain weakly dissipative property.