Comparison of Brownian jump and Brownian bridge resetting in search for Gaussian target on the line and in space

Abstract

Abstract We consider a Brownian searcher with diffusion coefficient D in d-dimensions, for d = 1 , 2 , 3 , that starts from the origin and searches for a random target with a centered-Gaussian distribution. The searcher is also equipped with a resetting mechanism that resets the searcher back to the origin. We consider three different resetting mechanisms. One is a Poissonian reset with rate r whereby at the reset time the searcher instantaneously jumps back to the origin, one is a periodic reset with period T whereby at the reset time the searcher instantaneously jumps back to the origin, and one is a Brownian bridge reset with period T, whereby the Brownian motion is conditioned to return to the origin at time T. Unlike the first two search processes, this last one has continuous paths. For d = 1 and d = 3, we obtain analytic formulas for the expected time to locate the random target, and minimize them over r or T, as the case may be. In one dimension, this expected time scales as σ 2 D , which is not surprising, but in three dimensions we obtain the anomalous scaling σ 3 D . We compare the relative efficiencies of the three search processes. In two dimensions, we show that the expected time scales as σ 2 D for the Poissonian reset mechanism.