大偏差与流体动力学极限

The project involves large deviations and their applications, hydrodynamic limits.for particle systems, entropy nequalities and exponential decay of Boltzman-shannon entropy. We prove that the Cram閞 functional of a Markov process can be controlled by a function of an.integral functional by H鰈der inequality under Orlicz norm if the transition semigroup is uniformly integrable. A large deviation principle for Markov processes is obtained under non-essential.irreducibility. We also extend the notation of F-Sobolev inequality to general Markov processes by replacing Dirichlet form with Donsker-Varadhan entropy and prove that the uniform integrability implies a F-Sobolev inequality. A deviation inequality under H鰈der norm for stochastic integral is obtained by GRR inequality. Large deviations under H鰈der norm and large deviations on capacity for stochastic flow are established. Moderate deviations for the maximum likelihood estimator and uniformly large deviations and uniformly moderate deviations for kernel density.estimators are obtained. Moderate deviation principle from hydrodynamic limit is first considered and the moderate deviation principle for exclusion processes is established. Entropy inequalities and exponential decay of Boltzmann-shannon entropy are researched. We give bounds on the.exponential decay rate of entropy in the random transposition model and Bernoulli-Laplace model which are independent of the number of sites and the number of particles. We obtain a proof of entropy inequality for Bernoulli measures by a Clark formula in discrete time and we also prove a bound that improves these inequalities of Bobkov and Ledoux. Limit theorems of some processesare considered.

研究无穷粒子系统的流体动力学极限及其大偏差,其意义在于通过微观模型导出宏观偏差分匠糖医馐推湎凳?扩散系数,或粘性系数)的物理意义;研究具有不连续统计小扰动问题的大?及随机过程.粒子系统的大偏差与中偏差;为研究动力系统.非线性偏差分方程.排队网络.非平衡统计力学等领域中的相关问题提供工具..