Abstract
A nonlocal convection-diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model parameters, the nonlocal steady and unsteady state master equations are shown to be well-posed in a weak sense. Then the nonlocal operator is shown to be the generator of finite-range nonsymmetric jump processes and, when certain conditions on the model parameters hold, the generators of finite and infinite activity Lévy and Lévy-type jump processes are shown to be special instances of the nonlocal operator.
Funding source: National Nuclear Security Administration
Award Identifier / Grant number: DE-NA-0003525
Funding source: Air Force Office of Scientific Research
Award Identifier / Grant number: FA9550-14-1-0073
Funding source: Army Research Office
Award Identifier / Grant number: W911NF-15-1-0562
Funding source: Defense Advanced Research Projects Agency
Award Identifier / Grant number: HR0011619523
Funding source: Advanced Scientific Computing Research
Award Identifier / Grant number: DE-SC0009324
Funding source: Division of Mathematical Sciences
Award Identifier / Grant number: DMS-1315259
Funding source: Oak Ridge National Laboratory
Award Identifier / Grant number: 1868-A017-15
Funding statement: The research of Marta D’Elia and Richard Lehoucq is supported by the Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration contract number DE-NA-0003525. The research of Qiand Du is supported in part by the U.S. Air Force Office of Scientific Research MURI Center for Material Failure Prediction Through Peridynamics grant number FA9550-14-1-0073, the U.S. Army Research Office grant W911NF-15-1-0562, and the U.S. Defense Advanced Research Projects Agency EQUIPS Program grant contract and award HR0011619523 and 1868-A017-15 through the Oak Ridge National Laboratory. The research of Max Gunzburger is supported in part by the U.S. Department of Energy Advanced Scientific Computing Research grant DE-SC0009324, the U.S. National Science Foundation Division of Mathematical Sciences grant DMS-1315259, and the U.S. Defense Advanced Research Projects Agency EQUIPS Program grant contract and award HR0011619523 and 1868-A017-15 through the Oak Ridge National Laboratory.
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