带切换扩散过程遍历性质及泛函不等式的研究

The regime-switching diffusions can be viewed as diffusion processes living in a random environment, which is characterized by continuous time jumping processes. This provides more realistic formulation for many applications, such as the price model of stocks in a market considering the bear and bull situation. The properties of this kind of processes are more complicated than those of the corresponding diffusions in each fixed environment. For instance, there are examples to show that the corresponding diffusions in each fixed environment are ergodic, but the regime-switching diffusion itself is transient. Our research will focus on the following topics: 1) to study the transient, recurrent, positive recurrent and strongly ergodic properties of the regime-switching diffusions, and provide some explicit conditions to justify these properties; 2) to study whether these processes satisfy some kinds of functional inequalities, such as transportation cost inequalities, Poincare inequalities, Harnack inequalities, and so on; 3) to study the properties of the invariant measures of these processes when the invariant measures exist. For instance, we want to know whether the density of the invariant measure exists. Moreover, we want to study the moments of the distributions of these processes and their invariant measures in order to give out more concrete characterization of these processes.

带切换的扩散过程可以看成是将单个扩散过程所描述的模型放在用连续时间的跳过程所刻画的随机环境中所形成的一个更复杂更符合实际情况的随机过程。比如考虑了熊市和牛市情形下的股票价格就可以用这种过程描述。该类过程的性质不是各个环境下扩散过程性质的简单叠加。在确定环境下都遍历的过程，经过切换，可以形成一个非常返的随机过程。我们针对带切换的扩散过程，一方面研究该类过程的非常返性、常返性、遍历性、强遍历的判别准则，给出显式的易于验证的判别条件；一方面研究该过程所能满足的泛函不等式，包括运费不等式，Poincare不等式, Harnack不等式等；还有我们也要研究这类过程的不变测度的性质，如密度的存在性, 描述其期望、方差等的矩性质，用以给出此类过程的具体刻画。

本项目研究在连续时间跳过程刻画的随机环境下，随机过程的常返性、遍历性、强Feller性、稳定性等的判别准则；研究此类过程的泛函不等式及收敛速度问题；研究过程的不变测度的刻画；研究此类过程在生物学、金融学方面的应用。通过三年的研究，a) 给出了此类过程常返性等的判别准则。此类准则一方面给出带切换扩散过程常返性等的精确刻画，另一方面，突破以往在有限状态空间上的跳过程的限制，可以研究在可数无限状态空间上的带切换扩散过程。b) 给出了一维带切换扩散过程不变测度的刻画，给出Euler-Maruyama逼近过程不变测度的存在性准则，证明出其收敛到原过程的不变测度。此外，还取得了泛函不等式，收敛速度等方面的研究成果。通过该项目，取得了一些新的结果，并且给出了新的研究方法，推动了关于带切换扩散过程研究的进一步发展。