| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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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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