Large time probability of failure in diffusive search with resetting for a random target in $\mathbb {R}^d$–A functional analytic approach

Abstract:

We consider a stochastic search model with resetting for an unknown stationary target $a\in \mathbb {R}^d,\ d\ge 1$, with known distribution $\mu$. The searcher begins at the origin and performs Brownian motion with diffusion coefficient $D$. The searcher is also equipped with an exponential clock with rate $r>0$, so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it continues its search anew from there. In dimension one, the target is considered located when the process hits the point $a$, while in dimensions two and higher, one chooses an $\epsilon _0>0$ and the target is considered located when the process hits the $\epsilon _0$-ball centered at $a$. Denote the position of the searcher at time $t$ by $X(t)$, let $\tau _a$ denote the time that a target at $a$ is located, and let $P^{d;(r,0)}_0$ denote probabilities for the process starting from 0. Taking a functional analytic point of view, and using the generator of the Markovian search process and its adjoint, we obtain precise estimates, with control on the dependence on $a$, for the asymptotic behavior of $P^{d;(r,0)}_0(\tau _a>t)$ for large time, and then use this to obtain large time estimates on $\int _{\mathbb {R}^d}P^{d;(r,0)}_0(\tau _a>t)d\mu (a)$, the probability that the searcher has failed up to time $t$ to locate the random target, for a variety of families of target distributions $\mu$. Specifically, for $B,l>0$ and $d\in \mathbb {N}$, let $\mu ^{(d)}_{B,l}\in \mathcal {P}(\mathbb {R}^d)$ denote any target distribution with density $\mu _{B,l}^{(d)}(a)$ that satisfies \begin{equation*} \lim _{|a|\to \infty }\frac {\log \mu _{B,l}^{(d)}(a)}{|a|^l}=-B. \end{equation*} Then we prove that \begin{equation*} \lim _{t\to \infty }\frac 1{(\log t)^l}\log \int _{\mathbb {R}^d} P_0^{d;(r,0)}(\tau _a>t)\mu _{B,l}^{(d)}(da)=-B(\frac D{2r})^\frac l2. \end{equation*} The result is independent of the dimension. In particular, for example, if the target distribution is a centered Gaussian of any dimension with variance $\sigma ^2$, then for any $\delta >0$, the probability of not locating the target by time $t$ falls in the interval $\big (e^{-(1+\delta )\frac {D}{4r\sigma ^2}(\log t)^2}, e^{-(1-\delta )\frac {D}{4r\sigma ^2}(\log t)^2}\big )$, for sufficiently large $t$.