Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes

TL;DR

This paper establishes precise large deviation principles for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes, revealing their asymptotic behaviors and relationships.

Contribution

It provides exact large deviation formulas and shows the proportional behavior of fractional Brownian motions and Riemann-Liouville processes in this context.

Findings

Large deviation principles for local times and intersection local times.

Law of iterated logarithm for local times derived.

Fractional Brownian motion and Riemann-Liouville processes behave similarly in large deviations.

Abstract

In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann-Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann-Liouville process.