Probabilistic Foundation of Nonlocal Diffusion with Volume Constraints

• Activity Number: 620
• Type: Contributed
• Date/Time: Thursday, August 2, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Nathanial Burch*+ and Richard Lehoucq
• Companies: SAMSI and Sandia National Laboratories
• Keywords: nonlocal diffusion ; master equation ; stochastic process ; anomalous diffusion

Abstract

The nonlocal diffusion equation is a model for anomalous diffusion that postulates nonlocal interactions between the field via a convolution with a propagator kernel. Such processes have been verified experimentally in applications such as contaminant flow in groundwater, dynamic motions in proteins, turbulence in fluids, and dynamics of financial markets. In analogy to the relationship between classical diffusion equation and Brownian motion, we show that the nonlocal diffusion equation is the master equation for a symmetric jump process. Of practical interest is often statistics of the process, e.g., exit-times from a bounded domain. Such exit-times of the process may be computed by enforcing absorbing boundary conditions on the process, which then induce so-called absorbing volume constraints on the master equation. Numerical solutions of the volume-constrained nonlocal diffusion equations are compared to density estimates from simulations of the underlying process on a bounded domain. The numerical solution of such master equations represents a powerful tool for computing statistics of symmetric jump processes on bounded domains while avoiding direct simulation.