Right now I'm interested in diffusion processes whose corresponding probability densities satisfy a Fokker-Planck equation with impulsive coefficients (for example, a drift term involving Dirac's delta function). The main questions are:

1) When one is given a second order differential equation with impulsive coefficients (that is, one that must be interpreted in the weak sense), what conditions must these coefficients satisfy in order for that equation to be the Fokker-Planck equation of a related stochastic process?

2) What is the precise relationship between second-order differential equations with impulsive terms and diffusions involving jump processes?

Any hints and/or references for this proble would be much appreciated.