 
Summary: The residence time equation
Noam Agmon *
The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
a r t i c l e i n f o
Article history:
Received 15 July 2010
In final form 15 August 2010
Available online 18 August 2010
a b s t r a c t
The residence time of a diffusing particle in a prescribed spatial region recently finds an increasing num
ber 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 FeynmanKac
approach. We demonstrate this for a ball in ddimensional space with an infinite observation time.
The present approach may be useful also for other Brownian functionals.
Ó 2010 Elsevier B.V. All rights reserved.
How much time (s) does a random walker spend in a prescribed
volume V in space, when monitored up to time t? This ``occupation
time" is a fundamental quantity in the mathematical theory of ran
