Optimizing the Drift in a Diffusive Search for a Random Stationary Target

Let $a\in\mathbb{R}$ denote an unknown stationary target with a known distribution $\mu\in\mathcal{P(\mathbb{R}})$, the space of probability measures on $\mathbb{R}$. A diffusive searcher $X(\cdot)$ sets out from the origin to locate the target. The time to locate the target is $T_a=\inf\{t\ge0: X(t)=a\}$. The searcher has a given constant diffusion rate $D>0$, but its drift $b$ can be set by the search designer from a natural admissible class $\mathcal{D}_\mu$ of drifts. Thus, the diffusive searcher is a Markov process generated by the operator $L=\frac D2\frac{d^2}{dx^2}+b(x)\frac d{dx}$. % equivalently, $X(\cdot)$ satisfies the stochastic differential equation %$X(t)=W(t)+\int_0^tb(X(s))ds$, where $W(\cdot)$ is a standard Brownian motion. For a given drift $b$, the expected time of the search is \begin{equation} \int_{\mathbb{R}} (E^{(b)}_0T_a)\thinspace\mu(da). \end{equation} Our aim is to minimize this expected search time over all admissible drifts $b\in\mathcal{D}_\mu$. For measures $\mu$ that satisfy a certain balance condition between their restriction to the positive axis and their restriction to the negative axis, a condition satisfied, in particular, by all symmetric measures, we can give a complete answer to the problem. We calculate the above infimum explicitly, we classify the measures for which the infimum is attained, and in the case that it is attained, we calculate the minimizing drift explicitly. For measures that do not satisfy the balance condition, we obtain partial results.