Research on diffusion processes and fuzzy set valued random variables

研究在一般空间中取值的随机变量可能是理论数学和应用数学的一个重要主题。本文研究的对象之一是模糊集值随机变量的极限定理。值得注意的是，目标空间与某些拓扑失去了可分性。本研究的结果之一是注意到对于模糊集合空间不可分的一致拓扑，大数定律、中心极限定理和鞅收敛定理都成立。通过证明过程中熵的可积性，我们采用了利用单调性的方法，并将其简化为经验分布理论，使网格更小。虽然大偏差原理对可分性更为敏感，但对于Levy度规诱导的拓扑，我们得到了Cramer型大偏差原理，对于Skorohod拓扑和均匀拓扑，我们也采用了一些强假设。我们给出了一个明确的例子来满足我们的假设。这似乎是互联网上出现预印本的反例。在自然假设下，得到了Sanov型大偏差原理。虽然我们只能在简单的情况下显式地计算速率函数，但其中之一是两个度量的相对熵。此外，与研究者H.Matsumoto一起，我们得到cM-X过程仅在c=0,1和2时是马尔可夫的，其中X是一维布朗运动，具有恒定的漂移，M是它的最大过程。这是Levy定理（对于c=1）和Pitman定理（对于c=2）的另一部分。

Study of random variables taking values in general spaces might be an important theme both for theoretical and applied mathematics. One of the objects of this research is to study limit theorems for fuzzy sets-valued random variables. It is worth to note that the target space looses separability with some topologies. One of the results of this research is having noticed that laws of large numbers, central limit theorems and martingale convergence theorems hold with respect to the uniform topology with which the fuzzy set space is not separable. We used the method exploiting monotone property and that reducing to the theory of empirical distribution by proving the integrability of the entropy in the procedure to let the mesh smaller.Although the large deviation principles are more sensitive to the separability, we obtained Cramer type large deviation principles for as far as the topology induced by Levy's metric, and also for Skorohod topology and uniform topology with a little strong assumptions. We gave an explicit example which satisfies our assumptions. This seems to be a counter example to a result in a preprint appearing in an internet. We also obtained Sanov type large deviation principles under a natural assumption. Although we can compute rate functions explicitly only in simple cases, one of them is a relative entropy of two measures.Also, with the investigator H.Matsumoto, we obtained that a cM-X process is Markov only when c=0,1,and 2,where X is a one-dimensional Brownnian motion with constant drift and M is its maximum process. This is another part of Levy's theorem(for c=1) and Pitman's theorem (for c=2).