Large Deviations for Riesz Potentials of Additive Processes

TL;DR

This paper investigates the large deviation behavior and laws of the iterated logarithm for additive functionals involving Riesz potentials of symmetric stable processes, providing new probabilistic insights into their asymptotic properties.

Contribution

It introduces novel large deviation principles and laws of the iterated logarithm for Riesz potentials of additive stable processes, extending existing theoretical frameworks.

Findings

Established large deviation results for the functionals.

Derived laws of the iterated logarithm for the processes.

Provided asymptotic estimates for the tail probabilities.

Abstract

We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^{t} | X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p(t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2$. We obtain results about the large deviations and laws of the iterated logarithm for $\zeta_{t}$.