|Hoping for Fittingly Preliminary Feedback
Nathanial Burch, Department of Mathematics, Colorado State University
Thursday, October 21, 2010
12:00 p.m., room 223, Weber Bldg
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.