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Nonlocal convection-diffusion volume-constrained problems and jump processes

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  • 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.
    Mathematics Subject Classification: Primary: 34B10; Secondary: 76R99, 45K05, 45A05.


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