Unmanned Aerial Vehicles (UAVs) can be an effective technology for security applications involving patrolling and search missions. Defining online patrolling strategies for UAVs presents challenges related both to classical patrolling, as periodic monitoring of the environment, and to search, as accurate localization and identification of the mission-related activities. In this paper, we deal with this problem considering probabilistic intrusions and a variable resolution sensing model that naturally applies to the domain of UAVs. We present three online single--robot patrolling strategies exploiting a variable resolution paradigm to represent the environment that has recently shown promising results for search problems. The approach uses a hierarchical representation based on probabilistic quadtrees that allows UAVs to tradeoff sensing accuracy with sensing area. The model is extended by adding stochastic arrivals of intruders in space and time. Obtained results validate this approach for online patrolling against approaches based on uniform grids.