Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

2012/05/22 by Fabienne Castell, Castell, Fabienne, L. Clément +3
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals #Theoretical and Computational Physics

paper · doi:10.48550/arxiv.1205.4917

Abstract

Let (X_t, t ≥ 0) be an α-stable random walk with values in \Z^d. Let l_t(x) = ∫₀^t δ_x(X_s) ds be its local time. For p>1, not necessarily integer, I_t = ∑_x l_t^p(x) is the so-called p-fold self- intersection local time of the random walk. When p(d -α) < d, we derive precise logarithmic asymptotics of the probability P(I_t ≥ r_t) for all scales r_t ≫ \E(I_t). Our result extends previous works by Chen, Li and Rosen 2005, Becker and König 2010, and Laurent 2012.

Related

• Moments and distribution of the local times of a transient random walk on \Z^d
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• Upper tails of self-intersection local times of random walks: survey of proof techniques