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We consider a stochastic search model with resetting for an unknownstationary target $a\in\mathbb{R}^d,\ d\ge1$, with known distribution $\mu$.The searcher begins at the origin and performs Brownian motion with diffusioncoefficient $D$. 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 clockrings, then its position is reset to the origin and it continues its searchanew from there. In dimension one, the target is considered located when theprocess 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 $P^{d;(r,0)}_0$ denoteprobabilities for the process starting from 0. Taking a functional analyticpoint of view, and using the generator of the Markovian search process and itsadjoint, we obtain precise estimates, uniformly in $a$, on the asymptoticbehavior of $P^{d;(r,0)}_0(\tau_a>t)$ for large time, and then use this toobtain large time estimates on$\int_{\mathbb{R}^d}P^{d;(r,0)}_0(\tau_a>t)d\mu(a)$, the probability that thesearcher has failed up to time $t$ to locate the random target, distributedaccording to $\mu$.

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