In this paper, a stochastic interpretation of nonlinear diffusion equations used for image filtering is proposed. This is achieved by relating the problem of evolving/smoothing images to that of tracking the transition probability density functions of an underlying random process. We show that such an interpretation of, e.g., Perona-Malik equation, in turn allows additional insight and sufficient flexibility to further investigate some outstanding problems of nonlinear diffusion techniques. In particular, upon unraveling the limitations as well as the advantages of such an equation, we are able to propose a new approach which is demonstrated to improve performance over existing approaches and, more importantly, to lift the longstanding problem of a stopping criterion for a nonlinear evolution equation with no data term constraint. Substantiating examples in image enhancement and segmentation are provided.