## The Annals of Probability

### Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

#### Abstract

We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite $p$th moment, for every $p<d$, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.

#### Article information

Source
Ann. Probab., Volume 42, Number 6 (2014), 2558-2594.

Dates
First available in Project Euclid: 30 September 2014

https://projecteuclid.org/euclid.aop/1412083632

Digital Object Identifier
doi:10.1214/13-AOP833

Mathematical Reviews number (MathSciNet)
MR3265174

Zentralblatt MATH identifier
1315.35019

#### Citation

Armstrong, Scott N.; Smart, Charles K. Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity. Ann. Probab. 42 (2014), no. 6, 2558--2594. doi:10.1214/13-AOP833. https://projecteuclid.org/euclid.aop/1412083632

#### References

• [1] Akcoglu, M. A. and Krengel, U. (1981). Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323 53–67.
• [2] Armstrong, S. N. (2009). Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations. J. Differential Equations 246 2958–2987.
• [3] Armstrong, S. N., Sirakov, B. and Smart, C. K. (2011). Fundamental solutions of homogeneous fully nonlinear elliptic equations. Comm. Pure Appl. Math. 64 737–777.
• [4] Becker, M. E. (1981). Multiparameter groups of measure-preserving transformations: A simple proof of Wiener’s ergodic theorem. Ann. Probab. 9 504–509.
• [5] Berger, N. and Deuschel, J.-D. (2014). A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment. Probab. Theory Related Fields 158 91–126.
• [6] Cabré, X. (1997). Nondivergent elliptic equations on manifolds with nonnegative curvature. Comm. Pure Appl. Math. 50 623–665.
• [7] Caffarelli, L. A. and Cabré, X. (1995). Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications 43. Amer. Math. Soc., Providence, RI.
• [8] Caffarelli, L. A. and Gutiérrez, C. E. (1997). Properties of the solutions of the linearized Monge–Ampère equation. Amer. J. Math. 119 423–465.
• [9] Caffarelli, L. A. and Souganidis, P. E. (2010). Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math. 180 301–360.
• [10] Caffarelli, L. A., Souganidis, P. E. and Wang, L. (2005). Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58 319–361.
• [11] Cherkaev, A. and Kohn, R., eds. (1997). Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications 31. Birkhäuser, Boston, MA.
• [12] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1–67.
• [13] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. Springer, New York.
• [14] Dal Maso, G. and Modica, L. (1986). Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. (4) 144 347–389.
• [15] Dal Maso, G. and Modica, L. (1986). Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368 28–42.
• [16] Dávila, G., Felmer, P. and Quaas, A. (2010). Harnack inequality for singular fully nonlinear operators and some existence results. Calc. Var. Partial Differential Equations 39 557–578.
• [17] Evans, L. C. (1992). Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120 245–265.
• [18] Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL.
• [19] Gloria, A. and Otto, F. (2011). An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 779–856.
• [20] Gloria, A. and Otto, F. (2012). An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 1–28.
• [21] Guo, X. and Zeitouni, O. (2012). Quenched invariance principle for random walks in balanced random environment. Probab. Theory Related Fields 152 207–230.
• [22] Gutiérrez, C. E. and Nguyen, T. (2011). Interior gradient estimates for solutions to the linearized Monge–Ampère equation. Adv. Math. 228 2034–2070.
• [23] Imbert, C. (2011). Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations. J. Differential Equations 250 1553–1574.
• [24] Imbert, C. and Silvestre, L. (2013). $C^{1,\alpha}$ regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233 196–206.
• [25] Kozlov, S. M. (1979). The averaging of random operators. Mat. Sb. 109 188–202, 327.
• [26] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120, 238.
• [27] Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979). Colloquia Mathematica Societatis János Bolyai 27 835–873. North-Holland, Amsterdam.
• [28] Papanicolaou, G. C. and Varadhan, S. R. S. (1982). Diffusions with random coefficients. In Statistics and Probability: Essays in Honor of C. R. Rao 547–552. North-Holland, Amsterdam.
• [29] Savin, O. (2007). Small perturbation solutions for elliptic equations. Comm. Partial Differential Equations 32 557–578.
• [30] Savin, O. (2009). Regularity of flat level sets in phase transitions. Ann. of Math. (2) 169 41–78.
• [31] Yurinskiĭ, V. V. (1986). Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh. 27 167–180, 215.