2021/07/18 by Ross G. Pinsky, Pinsky, Ross G.
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60J60 #Diffusion and Search Dynamics #FOS: Mathematics #Optimization and Search Problems #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2107.08475
openalex publication_date 2021/07/18 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28
We consider a stochastic search model with resetting for an unknown\nstationary target a\∈\ℝd, d\≥1, with known distribution \μ.\nThe searcher begins at the origin and performs Brownian motion with diffusion\ncoefficient D. The searcher is also armed with an exponential clock with rate\nr>0, so that if it has failed to locate the target by the time the clock\nrings, then its position is reset to the origin and it continues its search\nanew from there. In dimension one, the target is considered located when the\nprocess hits the point a, while in dimensions two and higher, one chooses an\n\ε0>0 and the target is considered located when the process hits the\n\ε0-ball centered at a.\n Denote the position of the searcher at time t by X(t), let \τa\ndenote the time that a target at a is located, and let Pd;(r,0)0 denote\nprobabilities for the process starting from 0. Taking a functional analytic\npoint of view, and using the generator of the Markovian search process and its\nadjoint, we obtain precise estimates, uniformly in a, on the asymptotic\nbehavior of Pd;(r,0)0(\τa>t) for large time, and then use this to\nobtain large time estimates on\n\∫\ℝdPd;(r,0)0(\τa>t)d\μ(a), the probability that the\nsearcher has failed up to time t to locate the random target, distributed\naccording to \μ.\n