| Budget Amount *help |
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
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| Research Abstract |
On the research of large deviations, to obtain the compact form of the entropy function or the free energy function is usually difficult problem, while it is very important. On the study of the range of random walks, we have similar difficult problem, and we try in this project to derive the property of the entropy function by considering the pinned random walk instead of the original random walk.
We can obtain the weak law of large numbers and the large deviation for the range of pinned random walks, which have the same statements as that for the range of the original random walk. However we cannot find further results in comparison with the previous results since we use the results for the original random walk to derive the new results for the pinned random walk. Therefore we will try to investigate this problem by considering known results for Brownian motion.
The volume of the Wiener sausage for Brownian bridge has been investigated, however the main subject is asymptotic behavior of its mean value and is not properties as a stochastic process. On the other hand, in three and six dimensional cases, it is very interesting that we can find the deference between pinned case and non-pinned case. However, since the method in the pinned case is operator theoretical, we can not find essential matters of the Brownian bridge as a stochastic process. We then investigate the expectation of the range of pinned random walks. It is very natural to consider whether the result for range of pinned random walk is the analog of that for the pinned Wiener sausage or not. In the three dimensional case, we can solve the problem affirmatively. However, it is still open in the six dimensional case.
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