The Limiting Shape for Drifted Internal Diffusion Limited Aggregation is a True Heat Ball

We build the iDLA cluster using drifted random walks, and study the limiting shapes they exhibit, with the help of sandpile models. For constant drift, the normalised cluster converges to a canonical shape S, which can be termed a true heat ball, in that it gives rise to a mean value property for caloric functions. The existence and boundedness of such a shape answers a natural yet open question in PDE theory.