  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 $a\neq0$.
