Large deviations for self-intersection local times of stable random walks

Abstract

Let $(X_t,t\ge 0)$ be a random walk on ${\mathbb{Z}}^d$. Let $l_T(x)={\int }_0^T{\delta }_x(X_s)ds$ the local time at the state $x$ and $I_T=\sum _{x\in {\mathbb{Z}}^d}{l}_T(x{)}^q$ the q-fold self-intersection local time (SILT). In \cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and Mörters obtain in \cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for the SILT with $q=2$. We extend these results to an $\alpha$-stable process (i.e. $\alpha \in ]0,2]$) in the case where $q(d-\alpha )\ge d$.

22 pages

References in corpus (1)

• Large deviations for renormalized self-intersection local times of stable processes