Action functionals for stochastic differential equations with L\'evy noise

TL;DR

This paper develops action functionals for stochastic differential equations driven by Lévy processes using large deviation theory, Legendre transforms, and Lévy symbols, to analyze their long-term behaviors.

Contribution

It introduces a novel method to derive action functionals for SDEs with Lévy noise, extending large deviation principles to these processes.

Findings

Derived explicit action functionals for scaled Brownian motion and Lévy processes.

Established long-term behavior analysis for SDEs with Lévy noise.

Utilized Legendre transforms and Lévy symbols in the derivation process.

Abstract

By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and L\'evy processes with existing finite exponential moments. Based on extended contraction principle, Legendre transform and L\'evy symbols, we derive the action functionals for stochastic differential equations driven by L\'evy processes.