The residence time equation

The residence time of a diffusing particle in a prescribed spatial region recently finds an increasing number of applications in physics and chemistry. A partial differential equation for the residence time moments is derived, as a generalization of the ordinary differential equation for the first passage time moments. When one seeks to calculate only the mean residence time and not its full distribution, this “residence time equation” constitutes a significant simplification over the conventional Feynman–Kac approach. We demonstrate this for a ball in d-dimensional space with an infinite observation time. The present approach may be useful also for other Brownian functionals.