Fokker-Planck-Kolmogorov equation for stochastic differential equations   with boundary hitting resets
  We consider a Markov process on a Riemannian manifold, which solves a stochastic differential equation in the interior of the manifold and jumps according to a deterministic reset map when it reaches the boundary. We derive a partial differential equation for the probability density function, involving a non-local boundary condition which accounts for the jumping behaviour of the process. This is a generalisation of the usual Fokker-Planck-Kolmogorov equation for diffusion processes. The result is illustrated with an example in the field of stochastic hybrid systems. 
Julien Bect, Hana Baili, Gilles Fleury