Anomalous Diffusion in Subsets of Euclidean Space

The subject of this dissertation is anomalous (nonlocal) diffusion, a type of random motion which is important in statistical physics, fluid dynamics, the study of animal foraging patterns, and economics. In the main contribution of the dissertation, we construct a sequence of L�vy flight models on discrete spaces, based on previously existing models from statistical physics and the study of networks, and then use the Trotter-Kato theorem from semigroup theory to show that they converge to a continuum limit. In one special case, this continuum limit is a scaling limit from which we recover the well-known fractional diffusion model in Rn. In the general case, we obtain a model which evolves by means of multiscale random jumps in an inhomogeneous medium, represented by some subset of Euclidean space. On the one hand, this convergence theorem provides insight into the nature of L�vy flight models on lattices and networks, allowing us to show that they can exhibit surprising boundary effects as they approach the continuum limit. On the other hand, the convergence theorem is also a numerical method for solving nonlocal analogues to the heat equation with no-flux boundary conditions. The method applies to a large class of models for which, apparently, no practical approximation method which was known to be convergent previously existed. In addition, some of these results are extended to cover related nonlinear problems, which leads to new existence-uniqueness theorems for nonlocal reaction-diffusion systems. This is accomplished by formulating a sequence of approximating equations for which comparison principles and well posedness can be proved, and then showing that these desiderata are preserved in the limit. All this facilitates the analysis of nonlinear systems representing predator-prey dynamics and the spatial spread of epidemics, two areas where there is evidence that anomalous diffusion plays an important role but takes place in a bounded, heterogeneous environment, giving rise to models which fit well into the theoretical framework developed here