Laplace approximations and related limit theorems

1.关于大偏差原理的拉普拉斯逼近问题首先，在自由能泛函的海森函数可能退化的情况下，得到了非对称马氏过程通常的拉普拉斯逼近阶的精确估计的一般框架。其次，对于大偏差原理的Laplace逼近的异序性问题，主要由T.Chiyonobu教授在I.I.D.情形下得到了新的极限定理。由随机矩阵的极限定理激励的适当条件下的随机变量。关于随机环境中随机过程的分布问题，首先，主要是教授。H.Tanaka和Y.Suzuki得到了具有单边Brown势的一维扩散过程的新的极限定理。其次，与临界现象的一些结果相关，得到了高桥随机游动的齐次化结果。另一方面，对于具有合适势的无限多个球相互反射的问题，主要是H.Tanemura教授通过SDE方法构造了这种无限多个球系统。在自相似过程的问题上，与M.Maejima教授得到了多维无穷可分分布及其投影的一些结果。我们还引入了算子半自相似过程，并与M.Maejima教授和T.Saigo教授一起得到了一些基本性质。

1. On the problem for Laplace approximateion for large deviation principle first of all, we got a general framework of the precise estimate of the usual Laplace approximation order for non-symmetric Markovian processes in the case that the Hessian of the free energy functional may be degenerate. Secondly, for the non-usual order problem of Laplace approximations for large deviation principle, the new type of limit theorem was obtained by mainly Prof. T. Chiyonobu in the case of I.I.d. random variables under suitable conditions motivated by the limit theorems of random matrices.2. On the problems for the distributions of stochastic processes in random environments, firstly, mainly Profs. H. Tanaka and Y. Suzuki got the new type of limit theorem for one-dimensional diffusion processes with one- sided Brownian potential. Secondly a homogenization result was obtained for a random walk on some kind of fraktals with H. Takahashi related to some results for critical phenomena. On the other hand, on the problem of infinitely many balls reflecting mutually and with suitable potentials, mainly Prof. H. Tanemura constructed such infinitely many balls system through SDE method.3. On the problems for self similar processes, we got some result for multidimensional infinitely divisible distributions and their projections with Prof. M. Maejima. We also introduced the operator semi-selfsimilar processes and got some basic properties with Prof. M. Maejima and T. Saigo.