Diffusive search with spatially dependent resetting

We consider a stochastic search model with resetting for an unknown stationary target a∈R with known distribution μ. 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=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 E0(r) denote expectations for the process X(⋅). The search ends at time Ta=inf{t≥0:X(t)=a}. The expected time of the search is then ∫R(E0(r)Ta)μ(da). Ideally, one would like to minimize this over all resetting rates r. We obtain quantitative growth rates for E0(r)Ta 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 E0(r)Ta is finite or infinite. We show generically that if r(x) is of the order |x|2l, with l>−1, then logE0(r)Ta is of the order |a|l+1; in particular, the smaller the asymptotic size of r, the smaller the asymptotic growth rate of E0(r)Ta. The asymptotic growth rate of E0(r)Ta continues to decrease when r(x)∼Dλx2 with λ>1; now the growth rate of E0(r)Ta is more or less of the order |a|1+1+8λ2. Note that this exponent increases to ∞ when λ increases to ∞ and decreases to 2 when λ decreases to 1. However, if λ=1, then E0(r)Ta=∞, for a≠0. Our results suggest that for many distributions μ supported on all of R, a near optimal (or optimal) choice of resetting function r in order to minimize ∫Rd(E0(r)Ta)μ(da) will be one which decays quadratically as Dλx2 for some λ>1. We also give explicit, albeit rather complicated, variational formulas for infr≩0∫Rd(E0(r)Ta)μ(da). For distributions μ with compact support, one should set r=∞ off of the support. We also discuss this case.