2016 Fiscal Year Final Research Report
On structure of random walk traces
| Project/Area Number |
25800064
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2013-04-01 – 2017-03-31
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| Keywords | ランダムウォーク / ブラウン運動 |
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| Outline of Final Research Achievements |
In this research, I got the following results. The first main result is to show the existence of the growth exponent of the loop-erased random walk (LERW) in three dimensions. LERW is a random simple path obtained by erasing all loops from a random walk path, which was introduced by Greg Lawler in 1980s. Both in math and physics literatures, it is known that the most difficult case is in three dimensions. I derived such existence of the exponent in three dimensions by establishing a number of delicates estimates on LERW. The second main result is to give a decomposition of a trace of the Brownian motion as follows. I showed that the union of the scaling limit of LERW and a suitable Poisson point process on a loop space is (in law) the Brownian motion. This decomposition can be seen as an analog of Ito's excursion theorem in one dimension.
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| Free Research Field |
確率論
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