Strongly chaotic systems (e.g., piecewise expanding mappings) exhibit diffusion-like behavior in the sense of central limit theorems. To find more precise statements about the similarity to probabilistic diffusion, we study how the evolution of probability densities under d-dimensional piecewise expanding mappings can be modeled by Markov processes with smooth transition probabilities (such as diffusion processes). Our results can be viewed as a special type of local limit theorem.