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With the ease of deployment, capabilities of evading the jammers and obscuring their existence, unmanned aerial vehicles (UAVs) are one of the most suitable candidates to perform surveillance. There exists a body of literature in which the inspectors follow a deterministic trajectory to conduct surveillance, which results in a predictable environment for malicious entities. Thus, introducing randomness to the surveillance is of particular interest. In this work, we propose a novel framework for stochastic UAV-assisted surveillance that i) inherently considers the battery constraints of the UAVs, ii) proposes random moving patterns modeled via random walks, and iii) adds another degree of randomness to the system via considering probabilistic inspections. We formulate the problem of interest, i.e., obtaining the energy-efficient random walk and inspection policies of the UAVs subject to probabilistic constraints on inspection criteria of the sites and battery consumption of the UAVs, which turns out to be signomial programming that is highly non-convex. To solve it, we propose a centralized and a distributed algorithm along with their performance guarantee. This work contributes to both UAV-assisted surveillance and classic random walk literature by designing random walks with random inspection policies on weighted graphs with energy limited random walkers.