Variance of residence time spent by diffusing particle in a sub-domain. Path integral based approach

Path integral based approach is used to analyze the variance of the residence time spent in a sub-domain by a particle diffusing in the presence of an arbitrary potential in a larger domain containing the sub-domain. It is assumed that there is no absorption of the particle within the domain or at its boundaries. Because of the ergodisity, the mean residence time in the sub-domain is a product of the observation time and the equilibrium probability of finding the particle in the sub-domain. We show that the variance also grows linearly with the observation time at large times and explain how the slope of this linear dependence can be found. The general approach is illustrated by simple examples, in which explicit formulas for the variance can be obtained.