This is a practice of (part of) my presentation for my preliminary examination later in the semester. As such, it will cover a substantial amount of ideas and research (many unfamiliar, yet accessible, to a statistics audience), but from a very "big-picture" perspective. Consequently, I think it should be an enjoyable talk.
We introduce the nonlocal diffusion equation as a generalized master equation for a Markovian continuous time random walk (CTRW). This is analogous to the relationship between a Wiener process and its associated Fokker-Planck equation, e.g., the classical diffusion equation. In fact, in a certain limit of vanishing nonlocality, the nonlocal diffusion equation reduces to the classical diffusion equation. A more general result establishes a relationship between nonlocal and fractional diffusion equations and also, consequently, compound Poisson processes and stable Levy processes. With the introduction of volume constraints on the solution of the nonlocal diffusion equation, we introduce so-called nonlocal boundary value problems, discuss properties of their solutions, and derive the induced boundary conditions for the underlying CTRWs on bounded domains. Evidently, these nonlocal boundary value problems are the generalized master equations for CTRWs on bounded domains with appropriate boundary conditions, which is demonstrated by comparing kernel density estimates from simulation data of the latter to numerical solutions, via a conforming finite element method, of the former. Applications, e.g., to anomalous diffusion and peridynamic heat conduction, are briefly discussed as well.
BYODrink, Pizza provided outside room at 11:30am, before start of talk.