马尔可夫过程的精确大偏差

Large deviation theory has a wide range of applications including information theory, risk management, statistical mechanics, thermodynamics and so on. This project aims to investigate precise large deviations for general Markov processes, based on what we have studied for the large deviations of some special stochastic processes. Precise large deviations find more applications in suitable functional equations and partial differential equations, besides the applications mentioned above. This project will study the solutions to some class of partial differential equations as an application of precise large deviations. So far, the precise large deviations and large deviation expansions are known only for Gaussian processes, and this project will provide results in this direction for many more stochastic processes.

大偏差理论有着广泛的应用：信息论(information theory)、风险管理(risk management)、统计力学(statistical mechanics)、 热力学(thermodynamics)等。本项目在我们前期对于大偏差一般理论及对某些特殊随机过程精确大偏差的研究基础上，拟进行对一般马尔可夫随机过程精确大偏差的研究。随机过程的精确大偏差理论除了上述的广泛应用之外，还可应用于某类泛函方程(functional equations)解的研究，一类偏微分方程(partial differential equations)解的渐进分析。作为马尔可夫过程精确大偏差结果的一个应用，本项目拟给出一大类偏微分方程的解的渐进研究。目前对于随机过程的精确大偏差及大偏差渐进展开的研究仅限于高斯过程，此项目对马尔可夫过程精确大偏差的研究将大范围推广完善了目前对精确大偏差的已有结果。

大偏差理论有着广泛的应用：信息论(information theory)、风险管理(risk management)、统计力学(statistical mechanics)、热力学(thermodynamics)等。本项目对一般马尔可夫随机过程精确大偏差进行了研究。随机过程的精确大偏差理论除了上述的广泛应用之外，还可应用于某类泛函方程(functional equations)解的研究，一类偏微分方程(partial differential equations)解的渐进分析。作为马尔可夫过程精确大偏差结果的一个应用，本项目给出了一大类偏微分方程的解的渐进研究。此项目对马尔可夫过程精确大偏差的研究大范围地推广完善了目前对精确大偏差的已有结果。