Research on diffusion processes and fuzzy set valued random variables

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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).