If a force is applied to a particle undergoing Brownian motion, the resulting motion has a state function which satisfies a diffusion or Schrödinger-type equation. We consider a process in which Brownian motion is replaced by a process which has Brownian transitions at all times other than random times at which the transitions have an additional "impulsive" displacement. Using a Feynman-Kac formulation based on generalized Riemann integration, we examine the resulting equation of motion.
"A Feynman-Kac Solution to a Random Impulsive Equation of Schrödinger Type." Real Anal. Exchange 36 (1) 107 - 148, 2010/2011.