Large deviation principle for the intersection measure of Brownian motions on unbounded domains

2020/05/19 by Mori, Takahiro
#60J65 (primary) 60F10 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2005.09219

Abstract

Consider the intersection measure ℓ^IS_t of p independent Brownian motions on ℝ^d. In this article, we prove the large deviation principle for the normalized intersection measure t^-pℓ^IS_t as t→ ∞, before exiting a (possibly unbounded) domain D⊂ℝ^d with smooth boundary. This is an extension of [W. König and C. Mukherjee: Communications on Pure and Applied Mathematics, 66(2):263--306, 2013] which deals with the case D is bounded. Our essential contribution is to prove the so-called super-exponential estimate for the intersection measure of killed Brownian motions on such D by an application of the Chapman-Kolmogorov relation.

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