Self-intersection local times of random walks: Exponential moments in subcritical dimensions

Details Self-intersection local times of random walks: Exponential moments in subcritical dimensions Self-intersection local times of random walks: Exponential moments in subcritical dimensions

2010-07-23

arxiv.org/abs/1007.4069v2

Mathematics

Probability

15 pages. To appear in Probability Theory and Related Fields. The final publication is available at springerlink.com

Scientific paper

10.1007/s00440-011-0377-0

Fix $p>1$, not necessarily integer, with $p(d-2)0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $\theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {\it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $\|\ell_t\|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_t\gg\E[\|\ell_t\|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $\ll t^{1/d}$ 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.

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