Diffusive Search with spatially dependent Resetting

Abstract: Consider a stochastic search model with resetting for an unknown stationary target $a\in\mathbb{R}$ with known distribution $\mu$. The searcher begins at the origin and performs Brownian motion with diffusion constant $D$. The searcher is also armed with an exponential clock with spatially dependent rate $r$, 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. Denote the position of the searcher at time $t$ by $X(t)$. Let $E_0^{(r)}$ denote expectations for the process $X(\cdot)$. The search ends at time $T_a=\inf{t\ge0:X(t)=a}$. The expected time of the search is then $\int_{\mathbb{R}}(E_0^{(r)}T_a)\thinspace\mu(da)$. Ideally, one would like to minimize this over all resetting rates $r$. We obtain quantitative growth rates for $E_0^{(r)}T_a$ as a function of $a$ in terms of the asymptotic behavior of the rate function $r$, and also a rather precise dichotomy on the asymptotic behavior of the resetting function $r$ to determine whether $E_0^{(r)}T_a$ is finite or infinite. We show generically that if $r(x)$ is on the order $|x|^{2l}$, with $l>-1$, then $\log E_0^{(r)}T_a$ is on the order $|a|^{l+1}$; in particular, the smaller the asymptotic size of $r$, the smaller the asymptotic growth rate of $E_0^{(r)}T_a$. The asymptotic growth rate of $E_0^{(r)}T_a$ continues to decrease when $r(x)\sim \frac{D\lambda}{x^2}$ with $\lambda>1$; now the growth rate of $E_0^{(r)}T_a$ is more or less on the order $|a|^{\frac{1+\sqrt{1+8\lambda}}2}$. However, if $\lambda=1$, then $E_0^{(r)}T_a=\infty$, for $a\neq0$.