Small ball probability of collision local time for symmetric stable processes

TL;DR

This paper derives the small ball probability for the collision local time of two independent symmetric alpha-stable processes, using contour integration to analyze the asymptotic behavior of their moment generating functions.

Contribution

It provides the first precise asymptotic estimates for small ball probabilities of collision local times in symmetric stable processes.

Findings

Derived asymptotic behavior of the moment generating function

Established small ball probability estimates for collision local time

Applied contour integration techniques

Abstract

In this article, the small ball probability is obtained for the collision local time of two independent symmetric $\alpha-$stable processes with parameters $\alpha_1,\alpha_2\in(0,2]$ satisfying $\max\{\alpha_1,\alpha_2\}>1$. The proof is based on obtaining the asymptotic behavior of moment generating function by contour integration.