Most probable transition paths in piecewise-smooth stochastic differential equations

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