Path Integral Approach to non-Markovian First-Passage Time Problems

TL;DR

This paper introduces a path integral method to compute first-passage time probabilities for non-Markovian stochastic processes, providing a perturbative approach to handle memory effects that complicate traditional solutions.

Contribution

It develops a novel path integral framework and perturbative technique for solving non-Markovian first-passage time problems, extending analytical tools beyond Markovian cases.

Findings

Path integral representation for non-Markovian first-passage problems

Perturbative evaluation method for the path integral

Applicable to complex stochastic systems with memory effects

Abstract

The computation of the probability of the first-passage time through a given threshold of a stochastic process is a classic problem that appears in many branches of physics. When the stochastic dynamics is markovian, the probability admits elegant analytic solutions derived from the Fokker-Planck equation with an absorbing boundary condition while, when the underlying dynamics is non-markovian, the equation for the probability becomes non-local due to the appearance of memory terms, and the problem becomes much harder to solve. We show that the computation of the probability distribution and of the first-passage time for non-Markovian processes can be mapped into the evaluation of a path-integral with boundaries, and we develop a technique for evaluating perturbatively this path integral, order by order in the non-Markovian terms.