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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations


Authors: Scott N. Armstrong, Pierre Cardaliaguet and Panagiotis E. Souganidis
Journal: J. Amer. Math. Soc. 27 (2014), 479-540
MSC (2010): Primary 35B27, 35F21, 60K35
Published electronically: January 27, 2014
MathSciNet review: 3164987
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Abstract: We present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level-set convex Hamilton-Jacobi equations in i.i.d. random environments, the first quantitative homogenization results for these equations in the stochastic setting. By taking advantage of a connection between the metric approach to homogenization and the theory of first-passage percolation, we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime (away from the flat spot of the effective Hamiltonian). In the sub-ballistic regime (on the flat spot), we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed, without further assumptions, at an arbitrarily slow rate. We identify a necessary and sufficient condition on the law of the Hamiltonian for an algebraic rate of convergence to hold in the sub-ballistic regime and show, under this hypothesis, that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization.

Our methods are novel and quite different from the techniques employed in the periodic setting, although we benefit from previous works in both first-passage percolation and homogenization. The link between the rate of homogenization and the flat spot of the effective Hamiltonian, which is related to the nonexistence of correctors, is a purely random phenomenon observed here for the first time.


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Additional Information

Scott N. Armstrong
Affiliation: Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: armstron@math.wisc.edu

Pierre Cardaliaguet
Affiliation: Ceremade (UMR CNRS 7534), Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris CEDEX 16, France
Email: cardaliaguet@ceremade.dauphine.fr

Panagiotis E. Souganidis
Affiliation: Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: souganidis@math.uchicago.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2014-00783-9
PII: S 0894-0347(2014)00783-9
Keywords: Stochastic homogenization, error estimate, convergence rate, Hamilton-Jacobi equation, first-passage percolation
Received by editor(s): June 13, 2012
Received by editor(s) in revised form: July 4, 2013
Published electronically: January 27, 2014
Additional Notes: The first author was partially supported by NSF Grant DMS-1004645.
The second author was partially supported by the French National Research Agency ANR-12-BS01-0008-01.
The third author was partially supported by NSF Grant DMS-0901802.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.