Anomalous diffusions and fractional order differential equations

Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells to the foraging behavior of animals and the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. I will then present some recent results in this area, in particular on the probabilistic representation to the solutions of time fractional equations with source terms. Two-sided estimates on the fundamental solutions will also be given when time permits.