Sometimes you shouldn’t reset in the bulk: Finding an optimal stochastic resetting profile
Alessandro Manacorda (ISC, Italy)
Finding a target for a purely Brownian searcher can be a hard task already in one dimension. Stochastic resetting qualitatively changes the game making the mean search time finite. An optimal homogeneous resetting rate minimizing the MFPT exists and can be found given the fixed distance between the resetting and the target positions. Less is known when the distance is not fixed in every trajectory but is drawn from a distribution, acting as a quenched disorder. Knowledge on the target distribution makes resetting desirable or not depending on the searcher’s position. We present an optimal solution for a heterogeneous resetting rate depending on the distribution of the target position. The optimal search strategy displays a transition from vanishing bulk resetting to a finite, space-dependent resetting profile; we characterize the instabilities of the former and find the optimal resetting profiles numerically for two families of target distributions. The solution found is non-trivial and opens further research questions in the field of optimal control with quenched disorder and in search processes with resetting.