Large deviations and renormalization for Riesz potentials of stable intersection measures

TL;DR

This paper investigates the large deviations and renormalization of Riesz potentials of stable intersection measures, providing new insights into their probabilistic behavior and applications to stable processes in random environments.

Contribution

It introduces a renormalized framework for Riesz potentials of stable processes and establishes large deviation principles and laws of the iterated logarithm for these measures.

Findings

Established large deviation principles for the renormalized intersection measures.

Derived laws of the iterated logarithm for the measures.

Applied results to stable processes in random potentials.

Abstract

We study the object formally defined as \gamma\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<\beta\le 2$ in $R^{d}$. When $\beta\le\sigma<\displaystyle\min \Big\{{3\over 2}\beta, d\Big\}$, this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for $\gamma$. This is applied to obtain results about stable processes in random potentials.