# American Institute of Mathematical Sciences

March  2014, 19(2): 373-389. doi: 10.3934/dcdsb.2014.19.373

## Nonlocal convection-diffusion volume-constrained problems and jump processes

 1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States 2 Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, United States

Received  June 2013 Revised  November 2013 Published  February 2014

We introduce the Cauchy and time-dependent volume-constrained problems associated with a linear nonlocal convection-diffusion equation. These problems are shown to be well-posed and correspond to conventional convection-diffusion equations as the region of nonlocality vanishes. The problems also share a number of features such as the maximum principle, conservation and dispersion relations, all of which are consistent with their corresponding local counterparts. Moreover, these problems are the master equations for a class of finite activity Lévy-type processes with nonsymmetric Lévy measure. Monte Carlo simulations and finite difference schemes are applied to these nonlocal problems, to show the effects of time, kernel, nonlocality and different volume-constraints.
Citation: Qiang Du, Zhan Huang, Richard B. Lehoucq. Nonlocal convection-diffusion volume-constrained problems and jump processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 373-389. doi: 10.3934/dcdsb.2014.19.373
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