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

Abstract

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_tggE[ 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.