title: Weak-noise limit of a piecewise-smooth stochastic differential equation.
abstract: We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided the singularity of the path integral associated with the nonsmooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behavior of the nonsmooth SDE in the low-noise limit. Finally, we consider a smooth regularization of the piecewise-constant SDE and study to what extent this regularization can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.
authors: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
authors: Yaming
authors: Y
authors: Chen
authors: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
authors: Adrian
authors: A
authors: Baule
authors: National Institute for Theoretical Physics, Stellenbosch 7600, South Africa and Institute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa.
authors: Hugo
authors: H
authors: Touchette
authors: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
authors: Wolfram
authors: W
authors: Just