Optimization of the residence time of a Brownian particle in a spherical subdomain

In this communication, we show that the residence time of a Brownian particle, defined as the cumulative time spent in a given region of space, can be optimized as a function of the diffusion coefficient. We discuss the relevance of this effect to several schematic experimental situations, classified in the nature -- random or deterministic -- both of the observation time and of the starting position of the Brownian particle.