We propose to consider linear and nonlinear image diffusion processes as the result of the transport of image values in external advection fields. Linear diffusion is seen as the continuous transport of particles in fields of random orientations, while non-linear behavior is obtained by modulating the magnitude of these fields according to local edge strength. Edge sharpening can be achieved by biasing fields towards image edges using e.g. Gradient Vector Flow. This point of view, closely related to discrete random walks and other stochastic interpretations of diffusion processes, allows for simple, stable and surprisingly efficient filters that can be implemented using a few lines of code only.