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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
MathSciNet review: 4024555
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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
MR Author ID: 885423
Email: nguillen@math.umass.edu

Chenchen Mou
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095
MR Author ID: 1125463
Email: muchenchen@math.ucla.edu

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

Keywords: Optimal transport, Lévy 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