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

HTML articles powered by AMS MathViewer

- by Scott N. Armstrong, Pierre Cardaliaguet and Panagiotis E. Souganidis PDF
- J. Amer. Math. Soc.
**27**(2014), 479-540 Request permission

## 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.

## References

- Kenneth S. Alexander,
*A note on some rates of convergence in first-passage percolation*, Ann. Appl. Probab.**3**(1993), no. 1, 81–90. MR**1202516** - Noga Alon and Joel H. Spencer,
*The probabilistic method*, 3rd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008. With an appendix on the life and work of Paul Erdős. MR**2437651**, DOI 10.1002/9780470277331 - Scott N. Armstrong and Panagiotis E. Souganidis,
*Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments*, J. Math. Pures Appl. (9)**97**(2012), no. 5, 460–504 (English, with English and French summaries). MR**2914944**, DOI 10.1016/j.matpur.2011.09.009 - S. N. Armstrong and P. E. Souganidis,
*Stochastic homogenization of level-set convex Hamilton-Jacobi equations*, Int. Math. Res. Not. , posted on (2013), 3420–3449., DOI Arxiv:1203.6303 [math.AP] - Kazuoki Azuma,
*Weighted sums of certain dependent random variables*, Tohoku Math. J. (2)**19**(1967), 357–367. MR**221571**, DOI 10.2748/tmj/1178243286 - Guy Barles,
*Solutions de viscosité des équations de Hamilton-Jacobi*, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17, Springer-Verlag, Paris, 1994 (French, with French summary). MR**1613876** - Itai Benjamini, Gil Kalai, and Oded Schramm,
*First passage percolation has sublinear distance variance*, Ann. Probab.**31**(2003), no. 4, 1970–1978. MR**2016607**, DOI 10.1214/aop/1068646373 - I. Capuzzo-Dolcetta and H. Ishii,
*On the rate of convergence in homogenization of Hamilton-Jacobi equations*, Indiana Univ. Math. J.**50**(2001), no. 3, 1113–1129. MR**1871349**, DOI 10.1512/iumj.2001.50.1933 - Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions,
*User’s guide to viscosity solutions of second order partial differential equations*, Bull. Amer. Math. Soc. (N.S.)**27**(1992), no. 1, 1–67. MR**1118699**, DOI 10.1090/S0273-0979-1992-00266-5 - Andrea Davini and Antonio Siconolfi,
*Metric techniques for convex stationary ergodic Hamiltonians*, Calc. Var. Partial Differential Equations**40**(2011), no. 3-4, 391–421. MR**2764912**, DOI 10.1007/s00526-010-0345-z - A. Davini and A. Siconolfi,
*Weak KAM Theory topics in the stationary ergodic setting*, Calc. Var. Part. Differ. Eq. (in press). - Lawrence C. Evans,
*The perturbed test function method for viscosity solutions of nonlinear PDE*, Proc. Roy. Soc. Edinburgh Sect. A**111**(1989), no. 3-4, 359–375. MR**1007533**, DOI 10.1017/S0308210500018631 - Lawrence C. Evans,
*Periodic homogenisation of certain fully nonlinear partial differential equations*, Proc. Roy. Soc. Edinburgh Sect. A**120**(1992), no. 3-4, 245–265. MR**1159184**, DOI 10.1017/S0308210500032121 - Branko Grünbaum,
*Convex polytopes*, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR**1976856**, DOI 10.1007/978-1-4613-0019-9 - J. M. Hammersley,
*Generalization of the fundamental theorem on sub-additive functions*, Proc. Cambridge Philos. Soc.**58**(1962), 235–238. MR**137800**, DOI 10.1017/s030500410003646x - J. M. Hammersley,
*Postulates for subadditive processes*, Ann. Probability**2**(1974), 652–680. MR**370721**, DOI 10.1214/aop/1176996611 - Hitoshi Ishii,
*Almost periodic homogenization of Hamilton-Jacobi equations*, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 600–605. MR**1870203** - Harry Kesten,
*On the speed of convergence in first-passage percolation*, Ann. Appl. Probab.**3**(1993), no. 2, 296–338. MR**1221154** - Elena Kosygina, Fraydoun Rezakhanlou, and S. R. S. Varadhan,
*Stochastic homogenization of Hamilton-Jacobi-Bellman equations*, Comm. Pure Appl. Math.**59**(2006), no. 10, 1489–1521. MR**2248897**, DOI 10.1002/cpa.20137 - Pierre-Louis Lions,
*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**667669** - P.-L. Lions, G. C. Papanicolaou, and S. R. S. Varadhan,
*Homogenization of Hamilton-Jacobi equations*, 1987. - Pierre-Louis Lions and Panagiotis E. Souganidis,
*Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting*, Comm. Pure Appl. Math.**56**(2003), no. 10, 1501–1524. MR**1988897**, DOI 10.1002/cpa.10101 - Pierre-Louis Lions and Panagiotis E. Souganidis,
*Homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media*, Comm. Partial Differential Equations**30**(2005), no. 1-3, 335–375. MR**2131058**, DOI 10.1081/PDE-200050077 - Pierre-Louis Lions and Panagiotis E. Souganidis,
*Stochastic homogenization of Hamilton-Jacobi and “viscous”-Hamilton-Jacobi equations with convex nonlinearities—revisited*, Commun. Math. Sci.**8**(2010), no. 2, 627–637. MR**2664465** - Songting Luo, Yifeng Yu, and Hongkai Zhao,
*A new approximation for effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations*, Multiscale Model. Simul.**9**(2011), no. 2, 711–734. MR**2818417**, DOI 10.1137/100799885 - I. Matic and J. Nolen,
*A sublinear variance bound for solutions of a random Hamilton-Jacobi equation*.*In preparation*. - Colin McDiarmid,
*On the method of bounded differences*, Surveys in combinatorics, 1989 (Norwich, 1989) London Math. Soc. Lecture Note Ser., vol. 141, Cambridge Univ. Press, Cambridge, 1989, pp. 148–188. MR**1036755** - James R. Munkres,
*Topology: a first course*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR**0464128** - Adam M. Oberman, Ryo Takei, and Alexander Vladimirsky,
*Homogenization of metric Hamilton-Jacobi equations*, Multiscale Model. Simul.**8**(2009), no. 1, 269–295. MR**2575055**, DOI 10.1137/080743019 - Fraydoun Rezakhanlou,
*Central limit theorem for stochastic Hamilton-Jacobi equations*, Comm. Math. Phys.**211**(2000), no. 2, 413–438. MR**1754523**, DOI 10.1007/s002200050820 - Fraydoun Rezakhanlou and James E. Tarver,
*Homogenization for stochastic Hamilton-Jacobi equations*, Arch. Ration. Mech. Anal.**151**(2000), no. 4, 277–309. MR**1756906**, DOI 10.1007/s002050050198 - Panagiotis E. Souganidis,
*Stochastic homogenization of Hamilton-Jacobi equations and some applications*, Asymptot. Anal.**20**(1999), no. 1, 1–11. MR**1697831** - Alain-Sol Sznitman,
*Distance fluctuations and Lyapounov exponents*, Ann. Probab.**24**(1996), no. 3, 1507–1530. MR**1411504**, DOI 10.1214/aop/1065725191 - Alain-Sol Sznitman,
*Brownian motion, obstacles and random media*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1717054**, DOI 10.1007/978-3-662-11281-6 - Michel Talagrand,
*Concentration of measure and isoperimetric inequalities in product spaces*, Inst. Hautes Études Sci. Publ. Math.**81**(1995), 73–205. MR**1361756** - Mario V. Wüthrich,
*Fluctuation results for Brownian motion in a Poissonian potential*, Ann. Inst. H. Poincaré Probab. Statist.**34**(1998), no. 3, 279–308 (English, with English and French summaries). MR**1625875**, DOI 10.1016/S0246-0203(98)80013-7 - Yu Zhang,
*On the concentration and the convergence rate with a moment condition in first passage percolation*, Stochastic Process. Appl.**120**(2010), no. 7, 1317–1341. MR**2639748**, DOI 10.1016/j.spa.2010.03.001

## 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
- MR Author ID: 323521
- 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
- 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. - © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**27**(2014), 479-540 - MSC (2010): Primary 35B27, 35F21, 60K35
- DOI: https://doi.org/10.1090/S0894-0347-2014-00783-9
- MathSciNet review: 3164987