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

Abstract : Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for all scales $r_t \gg \E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and König 2010, and Laurent 2012.
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Submitted on : Wednesday, July 31, 2013 - 3:28:21 PM
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Fabienne Castell, Clément Laurent, Clothilde Melot. Exponential moments of self-intersection local times of stable random walks in subcritical dimensions. Journal of the London Mathematical Society, London Mathematical Society ; Wiley, 2014, 89 (2), pp.876-902. ⟨10.1112/jlms/jdt065⟩. ⟨hal-00849628⟩

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