Large time probability of failure in diffusive search with resetting in arbitrary dimension--a functional analytic approach

We consider a stochastic search model with resetting for an unknown stationary target $a\in\mathbb{R}^d,\ d\ge1$, with known distribution $\mu$. The searcher begins at the origin and performs Brownian motion with diffusion coefficient $D$. The searcher is also armed 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, uniformly in $a$, on 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, distributed according to $\mu$.