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. 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.
Most probable transition paths in piecewise-smooth stochastic differential equations
Kaitlin Hill, Wake Forest UniversityAuthors: Kaitlin Hill, Jessica Zanetell, and John Gemmer
2022 AWM Research Symposium
Deterministic and Probabilistic Approaches for Nonlinear PDEs