| Project/Area Number |
14540113
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
KUMAGAI Takashi KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (90234509)
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| Co-Investigator(Kenkyū-buntansha) |
SHIGEKAWA Ichiro KYOTO UNIVERSITY, Graduate School of Science/Faculty of Science, Professor, 理学研究科, 教授 (00127234)
TAKAHASHI Yoichiro KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (20033889)
WATANABE Shinzo Ritsumeikan University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (90025297)
HINO Masanori KYOTO UNIVERSITY, Graduate School of Informatics, Associate Professor, 情報学研究科, 助教授 (40303888)
KIGAMI Jun KYOTO UNIVERSITY, Graduate School of Informatics, Professor, 情報学研究科, 教授 (90202035)
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| Project Period (FY) |
2002 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| Budget Amount *help |
¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2002: ¥2,900,000 (Direct Cost: ¥2,900,000)
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| Keywords | Disordered media / Heat kernel / Nash Inequality / Parabolic Harnack inequality / Besov space / Jump-type stochastic process / Homogenization / ソボレフ空間 / 拡散過程 / 無限次元空間 |
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| Research Abstract |
1.We have obtained new results for the problem to construct diffusion processes penetrating disordered media and to study properties of the processes when there are countable number of disordered media on a space. When domains of Dirichlet forms on each medium are Besov spaces, we apply the theory of Besov spaces and construct the penetrating diffusion process. We also obtain a short time asymptotic behavior of the heat kernel for the process and express the "most probable path" by means of the solution of variational formula of energy functions. This is a joint work with B.M.Hambly and appears in Probab. Theory Relat. Fields.
2.We have constructed jump-type processes on d-sets (fractal sets) and obtained detailed heat kernel estimates of the processes. The corresponding domains of Dirichlet forms are again Besov spaces and the trace formula of Besov spaces are very useful. Probabilistic techniques are used to obtain the off-diagonal estimates. Using the heat kernel estimates, we show transience/recurrence of the process and computed the Hausdorff dimension of the range of the process. This is a joint work with Z.Q.Chen and appears in Stoch. Proc. Their Appl.
3.We have studied the asymptotic behavior of the heat kernels for diffusion processes on self-similar sets. We prove an equivalence between the volume doubling property of the measure and some on-diagonal heat kernel estimates. Furthermore, we prove an equivalence between some estimates of escaping times plus local Nash inequalities and some type of heat kernel estimates from above.
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