×

Homogenization of a stochastically forced Hamilton-Jacobi equation. (English) Zbl 1465.60062

Summary: We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F21 Hamilton-Jacobi equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akcoglu, M. A.; Krengel, U., Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323, 53-67 (1981) · Zbl 0453.60039
[2] Armstrong, S.; Cardaliaguet, P., Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions, J. Eur. Math. Soc., 20, 4, 797-864 (2018) · Zbl 1392.35031
[3] Armstrong, S. N.; Cardaliaguet, P., Quantitative stochastic homogenization of viscous Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 40, 3, 540-600 (2015) · Zbl 1320.35032
[4] Armstrong, S. N.; Cardaliaguet, P.; Souganidis, P. E., Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations, J. Am. Math. Soc., 27, 2, 479-540 (2014) · Zbl 1286.35023
[5] Armstrong, S. N.; Souganidis, P. E., Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments, J. Math. Pures Appl. (9), 97, 5, 460-504 (2012) · Zbl 1246.35029
[6] Armstrong, S. N.; Souganidis, P. E., Stochastic homogenization of level-set convex Hamilton-Jacobi equations, Int. Math. Res. Not., 15, 3420-3449 (2013) · Zbl 1319.35003
[7] Armstrong, S. N.; Tran, H. V., Stochastic homogenization of viscous Hamilton-Jacobi equations and applications, Anal. PDE, 7, 8, 1969-2007 (2014) · Zbl 1320.35033
[8] Armstrong, S. N.; Tran, H. V.; Yu, Y., Stochastic homogenization of a nonconvex Hamilton-Jacobi equation, Calc. Var. Partial Differ. Equ., 54, 2, 1507-1524 (2015) · Zbl 1329.35042
[9] Armstrong, S. N.; Tran, H. V.; Yu, Y., Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension, J. Differ. Equ., 261, 5, 2702-2737 (2016) · Zbl 1342.35026
[10] Bakhtin, Y., The Burgers equation with Poisson random forcing, Ann. Probab., 41, 4, 2961-2989 (2013) · Zbl 1286.60099
[11] Bakhtin, Y., Ergodic theory of the Burgers equation, (Probability and Statistical Physics in St. Petersburg. Probability and Statistical Physics in St. Petersburg, Proc. Sympos. Pure Math., vol. 91 (2016), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-49 · Zbl 1388.60105
[12] Bakhtin, Y., Inviscid Burgers equation with random kick forcing in noncompact setting, Electron. J. Probab., 21, Article 37 pp. (2016) · Zbl 1338.37117
[13] Bakhtin, Y.; Cator, E.; Khanin, K., Space-time stationary solutions for the Burgers equation, J. Am. Math. Soc., 27, 1, 193-238 (2014) · Zbl 1296.37051
[14] Bakhtin, Y.; Khanin, K., On global solutions of the random Hamilton-Jacobi equations and the KPZ problem, Nonlinearity, 31, 4, R93-R121 (2018) · Zbl 1387.37073
[15] Becker, M. E., Multiparameter groups of measure-preserving transformations: a simple proof of Wiener’s ergodic theorem, Ann. Probab., 9, 3, 504-509 (1981) · Zbl 0468.28020
[16] Boritchev, A.; Khanin, K., On the hyperbolicity of minimizers for 1D random Lagrangian systems, Nonlinearity, 26, 1, 65-80 (2013) · Zbl 1263.35189
[17] Cannarsa, P.; Cardaliaguet, P., Regularity results for eikonal-type equations with nonsmooth coefficients, NoDEA Nonlinear Differ. Equ. Appl., 19, 6, 751-769 (2012) · Zbl 1254.49022
[18] Cardaliaguet, P., A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable, ESAIM Control Optim. Calc. Var., 15, 2, 367-376 (2009) · Zbl 1175.35030
[19] Cardaliaguet, P.; Nolen, J.; Souganidis, P. E., Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199, 2, 527-561 (2011) · Zbl 1294.35002
[20] Cardaliaguet, P.; Silvestre, L., Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side, Commun. Partial Differ. Equ., 37, 9, 1668-1688 (2012) · Zbl 1255.35066
[21] Cardaliaguet, P.; Souganidis, P. E., Homogenization and enhancement of the G-equation in random environments, Commun. Pure Appl. Math., 66, 10, 1582-1628 (2013) · Zbl 1284.60126
[22] Cardaliaguet, P.; Souganidis, P. E., On the existence of correctors for the stochastic homogenization of viscous Hamilton-Jacobi equations, C. R. Math. Acad. Sci. Paris, 355, 7, 786-794 (2017) · Zbl 1368.35025
[23] Ciomaga, A.; Souganidis, P. E.; Tran, H. V., Stochastic homogenization of interfaces moving with changing sign velocity, J. Differ. Equ., 258, 4, 1025-1057 (2015) · Zbl 1308.35024
[24] Crandall, M. G.; Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277, 1, 1-42 (1983) · Zbl 0599.35024
[25] Davini, A.; Kosygina, E., Homogenization of viscous and non-viscous HJ equations: a remark and an application, Calc. Var. Partial Differ. Equ., 56, 4, Article 95 pp. (2017) · Zbl 1382.35022
[26] Davini, A.; Siconolfi, A., Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case, Math. Ann., 345, 4, 749-782 (2009) · Zbl 1191.37044
[27] Dunlap, A.; Gu, Y.; Ryzhik, L.; Zeitouni, O., The random heat equation in dimensions three and higher: the homogenization viewpoint, preprint · Zbl 1481.35031
[28] E, W.; Khanin, K.; Mazel, A.; Sinai, Y., Invariant measures for Burgers equation with stochastic forcing, Ann. Math. (2), 151, 3, 877-960 (2000) · Zbl 0972.35196
[29] Feldman, W. M.; Fermanian, J.-B.; Ziliotto, B., An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity · Zbl 1459.35021
[30] Feldman, W. M.; Souganidis, P. E., Homogenization and non-homogenization of certain non-convex Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 108, 5, 751-782 (2017) · Zbl 1380.35014
[31] Gao, H., Random homogenization of coercive Hamilton-Jacobi equations in 1d, Calc. Var. Partial Differ. Equ., 55, 2, Article 30 pp. (2016) · Zbl 1343.35017
[32] Gu, Y.; Bal, G., Homogenization of parabolic equations with large time-dependent random potential, Stoch. Process. Appl., 125, 1, 91-115 (2015) · Zbl 1314.60127
[33] Gu, Y.; Ryzhik, L.; Zeitouni, O., The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher, Commun. Math. Phys., 363, 2, 351-388 (2018) · Zbl 1400.82131
[34] Gubinelli, M.; Imkeller, P.; Perkowski, N., Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3, Article e6 pp. (2015) · Zbl 1333.60149
[35] Hairer, M., Solving the KPZ equation, Ann. Math. (2), 178, 2, 559-664 (2013) · Zbl 1281.60060
[36] Hairer, M., A theory of regularity structures, Invent. Math., 198, 2, 269-504 (2014) · Zbl 1332.60093
[37] Hajej, A., Stochastic homogenization of a front propagation problem with unbounded velocity, J. Differ. Equ., 262, 7, 3805-3836 (2017) · Zbl 1364.35031
[38] Jing, W.; Souganidis, P. E.; Tran, H. V., Stochastic homogenization of viscous superquadratic Hamilton-Jacobi equations in dynamic random environment, Res. Math. Sci., 4, Article 6 pp. (2017) · Zbl 1358.35012
[39] Jing, W.; Souganidis, P. E.; Tran, H. V., Large time average of reachable sets and applications to homogenization of interfaces moving with oscillatory spatio-temporal velocity, Discrete Contin. Dyn. Syst., Ser. S, 11, 5, 915-939 (2018) · Zbl 1409.35025
[40] Khanin, K.; Zhang, K., Hyperbolicity of minimizers and regularity of viscosity solutions for a random Hamilton-Jacobi equation, Commun. Math. Phys., 355, 2, 803-837 (2017) · Zbl 1386.35503
[41] Kosygina, E.; Rezakhanlou, F.; Varadhan, S. R.S., Stochastic homogenization of Hamilton-Jacobi-Bellman equations, Commun. Pure Appl. Math., 59, 10, 1489-1521 (2006) · Zbl 1111.60055
[42] Kosygina, E.; Varadhan, S. R.S., Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium, Commun. Pure Appl. Math., 61, 6, 816-847 (2008) · Zbl 1144.35008
[43] Lions, P.-L., Generalized Solutions of Hamilton-Jacobi Equations, Research Notes in Mathematics, vol. 69 (1982), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, Mass.-London · Zbl 0497.35001
[44] Lions, P.-L.; Souganidis, P. E., Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris, Sér. I Math., 326, 9, 1085-1092 (1998) · Zbl 1002.60552
[45] Lions, P.-L.; Souganidis, P. E., Fully nonlinear stochastic partial differential equations: non-smooth equations and applications, C. R. Acad. Sci. Paris, Sér. I Math., 327, 8, 735-741 (1998) · Zbl 0924.35203
[46] Lions, P.-L.; Souganidis, P. E., Fully nonlinear stochastic pde with semilinear stochastic dependence, C. R. Acad. Sci. Paris, Sér. I Math., 331, 8, 617-624 (2000) · Zbl 0966.60058
[47] Lions, P.-L.; Souganidis, P. E., Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris, Sér. I Math., 331, 10, 783-790 (2000) · Zbl 0970.60072
[48] Lions, P.-L.; Souganidis, P. E., Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting, Commun. Pure Appl. Math., 56, 10, 1501-1524 (2003) · Zbl 1050.35012
[49] Lions, P.-L.; Souganidis, P. E., Homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media, Commun. Partial Differ. Equ., 30, 1-3, 335-375 (2005) · Zbl 1065.35047
[50] Lions, P.-L.; Souganidis, P. E., Stochastic homogenization of Hamilton-Jacobi and “viscous”-Hamilton-Jacobi equations with convex nonlinearities—revisited, Commun. Math. Sci., 8, 2, 627-637 (2010) · Zbl 1197.35031
[51] Mukherjee, C.; Shamov, A.; Zeitouni, O., Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \(d \geq 3\), Electron. Commun. Probab., 21, Article 61 pp. (2016) · Zbl 1348.60094
[52] Nolen, J.; Novikov, A., Homogenization of the G-equation with incompressible random drift in two dimensions, Commun. Math. Sci., 9, 2, 561-582 (2011) · Zbl 1241.35021
[53] Rezakhanlou, F.; Tarver, J. E., Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151, 4, 277-309 (2000) · Zbl 0954.35022
[54] Schwab, R. W., Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media, Indiana Univ. Math. J., 58, 2, 537-581 (2009) · Zbl 1180.35082
[55] Seeger, B., Homogenization of pathwise Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 110, 1-31 (2018) · Zbl 1382.35028
[56] Seeger, B., Scaling limits and homogenization of mixing Hamilton-Jacobi equations, Commun. Partial Differ. Equ. (2020), in press
[57] Souganidis, P. E., Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20, 1, 1-11 (1999) · Zbl 0935.35008
[58] Souganidis, P. E., Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, (Singular Random Dynamics. Singular Random Dynamics, Lecture Notes in Math., vol. 2253 (2019), Springer: Springer Cham), 75-220 · Zbl 1498.60399
[59] Stroock, D. W.; Varadhan, S. R.S., Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, vol. 233 (1979), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0426.60069
[60] Ziliotto, B., Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample, Commun. Pure Appl. Math., 70, 9, 1798-1809 (2017) · Zbl 1382.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.