Laplace approximations and related limit theorems

1.关于大偏差原理的拉普拉斯逼近问题，首先给出了在自由能泛函的Hessian可能退化的情况下，非对称马氏过程的通常拉普拉斯逼近阶的精确估计的一般框架。其次，对于大偏差原理的拉普拉斯逼近的非通常阶问题，T. I.I.D.一案中的千代信随机变量在适当的条件下由随机矩阵的极限定理激发.关于随机环境中随机过程的分布问题，首先，主要是Prof. H. Tanaka和Y. Suzuki得到了一类具有单侧布朗势的一维扩散过程的新的极限定理。其次，得到了一类分形上随机游动的均匀化结果。Takahashi给出了临界现象的一些结果。另一方面，关于无穷多个具有适当位势的球相互反射问题，主要是H。Tanemura通过迭代法构造了这样的无穷多球系统.在自相似过程的问题上，我们得到了M。前岛我们还引入了算子半自相似过程，并与M。Maejima和T.西乡

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.