Numerical analysis of degenerate Kolmogorov equations of constrained stochastic Hamiltonian systems

In this work, we propose a method to compute numerical approximations of the invariant measures and Rice's formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in Mertz et al. (2018). A functional analysis framework is presented; regularization and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.