We shall assume that the motion of a target in a given region is a diffusion process and that a searcher tries to find the target. If the searcher is able to remember the track he has travelled, then the probability density--from the searcher's point of view--of the target will be changing also because of the search, since the searcher knows that the target was not in the immediate proximity of his track. In the case of a stationary target the search already carried out would be taken into account by a straightforward application of Bayes' formula [4]. We shall assume the customary form for the instantaneous probability density of detection [3] and then modify the diffusion equation of the target to include the above Bayesian effect. The properties of this equation are discussed. An optimal statistical control problem of finding the best way of search is formulated.