Nonlocal Convection-Diffusion Problems on Bounded Domains and Finite-Range Jump Processes

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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 Information
  • National Nuclear Security Administration (DE-NA-0003525)
  • Air Force Office of Scientific Research (FA9550-14-1-0073)
  • Army Research Office (W911NF-15-1-0562)
  • Defense Advanced Research Projects Agency (HR0011619523)
  • Advanced Scientific Computing Research (DE-SC0009324)
  • Division of Mathematical Sciences (DMS-1315259)
  • Oak Ridge National Laboratory (1868-A017-15)

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