| In this thesis,we investigate the long time asymptotics of the regime-switching stochastic differential equations and the distribution dependent stochastic differential equations,and establish the moderate deviation principle for the unbounded additive functionals of these two types of stochastic differential equations respectively.Firstly,we establish the Wang’s Harnack inequality for the regime-switching model,and then using this inequality to characterise the long time asymptotics of the unbounded additive functionals.Next,for the distribution dependent stochastic differential equations,we prove that the unbounded additive functionals satisfy the moderate deviation principle using the exponentially equivalent property of the large deviation theory.The remainder of the paper is organized as follows.In the first part,we outline the background of the research,recall the development and the applications of the large(moderate)deviation principle,and simply introduce the theory of the regime-switching diffusion processes and the distribution dependent stochastic differential equations,and relate the main results of this thesis.In the second part,we recall the preliminary for the large(moderate)deviation principle and the dimension-free Harnack inequality.In the third part,we provide explicit conditions to establish the dimension-free Harnack inequalities respectively for state-independent and state-dependent regimeswitching processes.Then such Harnack inequalities are applied to establish the hypercontractivity of the associated semigroup,and further to study the long time asymptotics of unbounded additive functionals of regime-switching processes.Much effort is devoted to overcoming the difficulty caused by the close interaction between the diffusion processes and the Markov chains in the setting of regime-switching processes.In the forth part,by comparing the original equations with the corresponding stationary ones,the moderate deviation principle is established for unbounded additive functionals of the distribution dependent stochastic differential equations,we consider both the non-degenerate and degenerate noises.We overcome the difficulty caused by the nonlinear systems using the exponentially equivalent property. |
