Transition Path Theory For L\'{e}vy-Type Processes: SDE Representation and Statistics

TL;DR

This paper develops a Transition Path Theory for Lévy-type processes, deriving an SDE representation for transition paths and analyzing their statistical properties in non-Gaussian stochastic systems.

Contribution

It provides the first rigorous SDE representation for transition paths in Lévy-type processes, filling a key gap in non-Gaussian stochastic transition analysis.

Findings

Derived the SDE representation for transition path processes.

Proved well-posedness of the SDE representation.

Analyzed statistical properties like distribution and probability current.

Abstract

This paper establishes a Transition Path Theory (TPT) for L\'{e}vy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the rigorous derivation of the stochastic differential equation (SDE) representation for transition path processes, which share the same distributional properties as transition trajectories, along with a proof of its well-posedness. This result provides a solid theoretical foundation for sampling transition trajectories. The paper also investigates the statistical properties of transition trajectories, including their probability distribution, probability current, and rate of occurrence.