Large Deviation and Random Media

研究了紧致黎曼流形上可微1-形式的随机线积分。关于随机线积分产生的电流的大偏差原理的文献很少。然后，我们首先对随机线积分产生的电流的大偏差原理进行了显式的表述。为此，我们考虑了紧致黎曼流形上布朗运动的一条路径所产生的电流与经验测度对的大偏差原理。通过考虑这一对，我们可以清楚地表达大偏差原理的速率函数。此外，我们还得到了大偏差原理的速率函数的显式表示，并研究了考虑与随机线积分相关的大偏差原理的电流空间的拓扑结构。在以往的文献中，电流的拓扑是由平方可积的拓扑给出的。这种拓扑图的形式取决于黎曼流形的维度。利用新的估计，我们可以得到一个与流形的维度无关的由随机线积分产生的电流空间的拓扑图。我们还研究了多维布朗随机环境和反射布朗随机环境中扩散过程的常返性和暂时性。利用Wiener-Hopf型路径分解公式，我们得到了半空间中多维一般Levy过程的格林函数的路径分解的一个表示式。应用这个公式，我们得到了半空间中旋转不变吸收稳定过程的格林函数的显式公式。

We investigated on the stochastic line integrals of differentiable 1-forms on a compact Riemannian manifold. There were only few papers on the large deviation principle for the currents generated by the stochastic line integrals. Then, first we aimed to formulate the large deviation principle for currents generated by stochastic line integrals in an explicit way. For this purpose, we considered the large deviation principle of the pair of the current and the empirical measure generated by a path of the Brownian motion on the compact Riemannian manifold. By considering this pair ; we could have a clear expression of the rate function of the large deviation principle. Furthermore, we succeeded to get the explicit representation of the rate function of the large deviation principle.We also investigated the topology of the space of currents in which the large deviation principle related to the stochastic line integrals was considered. In former papers, the topology of currents was given by the topology of square integrability. This topology was given in the form depending on the dimension of the Riemannian manifold. By using new estimates, we could get a topology of the space of currents generated of stochastic line integrals, which is independent of the dimension of the manifold.We also investigated the recurrence and transience of the diffusion processes in multi-dimensional Brownian and reflected Brownian random environments.We also got the following result related to the multi-dimensional random environments. By using our Wiener-Hopf type path decomposition formula, we got a representation formula of a path decomposition of the Green function of the multi-dimensional general Levy processes in the half space. Applying this formula, we got the explicit formula of the Green functions of the rotation invariant absorbing stable processes in the half space.