2012 Fiscal Year Final Research Report
Extensions of reflecting Markov processes on unbounded domains
| Project/Area Number |
22540125
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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 | Osaka University |
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Principal Investigator |
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| Project Period (FY) |
2010 – 2012
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| Keywords | 無限錐 / 反射壁ブラウン運動 / 多重連結領域 / 小松・レブナー微分方程式 / かがりブラウン運動 / SKLE / 1次元極小拡散過程 / 対称性 / 対称拡散拡張 |
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| Research Abstract |
1. The time changed reflecting Brownian motion on a 3-space consisting of two infinite cones is shown to admit 4 and only 4 possible symmetric Markovian extensions.
2. The Komatu-Loewner differential equation for a multiply connected planar domain is derived by the Brownian motion with darning (BMD). The notion of stochastic Loewner evolution (SLE) for simply connected domains is extended to multiply connected domains as stochastic Komatu-Loewner evolution (SKLE).
3. The minimal diffusion on an interval is shown to be symmetric for the canonical measure and all its symmetric diffusion extensions are identified in relation to Ito-McKean’ s theory. As a result, the one-dimensional diffusions with general boundary conditions can be directly constructed by means of regular Dirichlet forms.
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