State dependent diffusion in a bistable potential: conditional probabilities and escape rates

TL;DR

This paper analyzes a bistable system with state-dependent noise, deriving escape rates and probabilities using path integrals, and confirms findings with numerical simulations, revealing corrections to classical escape rate formulas.

Contribution

It introduces a path integral approach for state-dependent diffusion systems and computes escape rates with corrections to Kramers' formula, applicable for arbitrary stochastic prescriptions.

Findings

Derived a path integral representation for the system.

Computed escape rates with diffusion-dependent corrections.

Validated theoretical results with numerical simulations.

Abstract

We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similarly to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers' escape rate produced by the diffusion function which governs the state dependent diffusion for arbitrary values of the stochastic prescription parameter. Theoretical results are confirmed through numerical simulations.