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Transactions of the American Mathematical Society

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Coupling Lévy measures and comparison principles for viscosity solutions


Authors: Nestor Guillen, Chenchen Mou and Andrzej Świȩch
Journal: Trans. Amer. Math. Soc. 372 (2019), 7327-7370
MSC (2010): Primary 35D40, 35J60, 35R09, 45K05, 47G20
DOI: https://doi.org/10.1090/tran/7877
Published electronically: August 28, 2019
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Abstract: We prove new comparison principles for viscosity solutions of nonlinear integro-differential equations. The operators to which the method applies include but are not limited to those of Lévy-Itô type. The main idea is to use an optimal transport map to couple two different Lévy measures and use the resulting coupling in a doubling of variables argument.


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Additional Information

Nestor Guillen
Affiliation: Department of Mathematics and Statistics, UMass, Amherst, Massachusetts 01003
Email: nguillen@math.umass.edu

Chenchen Mou
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095
Email: muchenchen@math.ucla.edu

Andrzej Świȩch
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: swiech@math.gatech.edu

DOI: https://doi.org/10.1090/tran/7877
Keywords: Optimal transport, L\'evy measures, nonlocal equations, viscosity solutions, comparison principles, uniqueness.
Received by editor(s): July 12, 2018
Received by editor(s) in revised form: April 1, 2019, and April 16, 2019
Published electronically: August 28, 2019
Article copyright: © Copyright 2019 American Mathematical Society