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

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

2009-05-04

arxiv.org/abs/0905.0376v1

Physics

Condensed Matter

Statistical Mechanics

5 pages, 1 figure

Scientific paper

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.

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