Large deviations for sticky Brownian motions
Large deviations for sticky Brownian motions
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DOI：
10.1214/20-ejp515
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发表时间：
2019-05
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通讯作者：
Guillaume Barraquand;M. Rychnovsky
中科院分区：
文献类型：
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作者：
Guillaume Barraquand;M. Rychnovsky
我们考虑n点粘性布朗运动：一组n个扩散，当它们分开时演变为独立的布朗运动，并在局部相互作用，使符合时间集具有正的勒贝格测量和正概率。这些扩散也可以看作是随机环境中的n个随机运动，其分布由所谓的随机核流给出。对于一类特殊的粘性相互作用，我们证明了描述随机流的精确公式，并证明了在大偏差区，这些随机流的随机波动是tracey - widom GUE分布的。该结果的等效公式表明，n个粘性布朗运动中的极端粒子在大n和大时间限制下具有特雷西-威登分布起伏。通过将粘性布朗运动视为随机环境中精确可解的随机游走的极限（以前已知的），证明了这些结果。
We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.