Self-intersection local times of random walks: exponential moments in subcritical dimensions
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- 作者：Mathias Becker (1)
Wolfgang K?nig (1) (2) - 关键词：Self ; intersection local time ; Upper tail ; Donsker–Varadhan large deviations ; Variational formula ; Gagliardo–Nirenberg inequality ; 60K37 ; 60F10 ; 60J55
- 刊名：Probability Theory and Related Fields
- 出版年：2012
- 出版时间：4 - December 2012
- 年：2012
- 卷：154
- 期：3
- 页码：585-605
- 全文大小：272KB
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Wolfgang K?nig (1) (2)
1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany
2. Technical University Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany - ISSN：1432-2064
文摘
Fix p?>?1, not necessarily integer, with p(d ?2)?< d. We study the p-fold self-intersection local time of a simple random walk on the lattice ${\mathbb Z^d}$ up to time t. This is the p-norm of the vector of the walker’s local times, ?/em> t . We derive precise logarithmic asymptotics of the expectation of exp{θ t ||?/em> t || p } for scales θ t >?0 that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of t and θ t , and the precise rate is characterized in terms of a variational formula, which is in close connection to the Gagliardo–Nirenberg inequality. As a corollary, we obtain a large-deviation principle for ||?/em> t || p /(tr t ) for deviation functions r t satisfying ${t r_t\gg \mathbb E[||\ell_t||_p]}$ . Informally, it turns out that the random walk homogeneously squeezes in a t-dependent box with diameter of order ?t 1/d to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.