Most probable transition paths in piecewise-smooth stochastic differential equations

Abstract

We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an $n-$dimensional system with a switching manifold in the drift that forms an $(n-1)-$dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use $\Gamma -$convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin-Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

arXiv comment38 pages, 9 figures

References in corpus (24)

• Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
• Nonlinear threshold behavior during the loss of Arctic sea ice
• Asymptotic Exit Location Distributions in the Stochastic Exit Problem
• When very slow is too fast -- collapse of a predator-prey system
• Climate tipping as a noisy bifurcation: a predictive technique
• Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations
• A primer on noise-induced transitions in applied dynamical systems
• Prey switching with a linear preference trade-off
• Regularization of hidden dynamics in piecewise smooth flows
• An iterative action minimizing method for computing optimal paths in stochastic dynamical systems
• An adaptive Euler-Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis
• Stick-slip motion of solids with dry friction subject to random vibrations and an external field
• Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System
• The tipping times in an Arctic sea ice system under influence of extreme events
• Weak-noise limit of a piecewise-smooth stochastic differential equation
• Γ-convergence of Onsager-Machlup functionals. Part I: With applications to maximum a posteriori estimation in Bayesian inverse problems
• Gamma Limit for Transition Paths of Maximal Probability
• Path properties of atmospheric transitions: illustration with a low-order sudden stratospheric warming model
• Stochastic dynamics of Arctic sea ice Part I: Additive noise
• Analysis of an Arctic sea ice loss model in the limit of a discontinuous albedo
• On large deviation properties of Brownian motion with dry friction
• Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces
• Rare events in networks with internal and external noise
• String Method for Generalized Gradient Flows: Computation of Rare Events in Reversible Stochastic Processes