Uniformly-Optimal Strategies in Search Problems

Let $f(x)$ be the density function of the a priori distribution of a particle in $R^n$. The strategy of search is defined by a function $\alpha=\alpha(x,t)\geqq 0$, $\int_{R^n}\alpha(x,t)\,dx=1$. The probability of finding the particle at a point $x$ during time $t$, under the condition that it is there, using the strategy $\alpha$, is given by the functional II $(\int_0^t\alpha(x,t)\,dt,x)$. Let $P_\alpha(T)$ be the probability of finding the particle using the strategy $\alpha$ during the time $T$. A strategy $\alpha^*$ is uniformly optimal if $P_{\alpha^*}(T)=\sup\limits_\alpha P_\alpha (T)$ for any $T>0$. In a very general case we prove the existence of the strategy $\alpha^*$ and find its explicit form.