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

Armstrong, Scott; Cardaliaguet, Pierre; Souganidis, Panagiotis

doi:10.1090/S0894-0347-2014-00783-9

Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
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by Scott N. Armstrong, Pierre Cardaliaguet and Panagiotis E. Souganidis

J. Amer. Math. Soc. 27 (2014), 479-540

DOI: https://doi.org/10.1090/S0894-0347-2014-00783-9

Published electronically: January 27, 2014

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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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