Mixing Local and Nonlocal Evolution Equations

In this paper, we study the homogenization of a stochastic process and its associated evolution equations in which we mix a local part (given by a Brownian motion with a reflection on the boundary) and a nonlocal part (given by a jump process with a smooth kernel). We consider a sequence of partitio...

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Bibliographic Details
Published in:Mediterranean journal of mathematics 2023-04, Vol.20 (2), Article 59
Main Authors: Capanna, Monia, Rossi, Julio D.
Format: Article
Language:English
Subjects:
Brownian motion
Chemical partition
Convergence
Density
Domains
Evolution
Mathematical analysis
Mathematics
Mathematics and Statistics
Stochastic processes
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Summary:In this paper, we study the homogenization of a stochastic process and its associated evolution equations in which we mix a local part (given by a Brownian motion with a reflection on the boundary) and a nonlocal part (given by a jump process with a smooth kernel). We consider a sequence of partitions of the (fixed) spacial domain into two parts (local and nonlocal) that are mixed in such a way that they both have positive density at every point in the limit. Under adequate hypotheses on the sequence of partitions, we prove convergence of the associated densities (that are solutions to an evolution equation with coupled local and nonlocal parts in two different regions of the domain) to the unique solution to a limit evolution system in which the local part disappears and the nonlocal part survives but divided into two different components. We also obtain convergence in distributions of the processes associated with the partitions and prove that the limit process has a density pair that coincides with the limit of the densities.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-023-02263-y