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

TL;DR

This paper extends large deviations results for self-intersection local times from simple random walks to symmetric $oldsymbol{ extit{ ext{alpha}}}$-stable processes in the supercritical regime, covering new parameter ranges.

Contribution

It generalizes existing large deviations principles for self-intersection local times to $oldsymbol{ extit{ ext{alpha}}}$-stable processes with broader conditions on $q$ and $d$.

Findings

Established large deviations principles for $oldsymbol{ extit{ ext{alpha}}}$-stable processes.

Extended the supercritical case results to a wider class of processes.

Unified the theory for different types of random walks and stable processes.

Abstract

Let $(X_t,t\geq 0)$ 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\limits_{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 M\"orters 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 $q(d-\alpha)\geq d$.