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Error estimates and convergence rates for the stochastic homogenization of HamiltonJacobi equations
 J. AMER. MATH. SOC
, 2013
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A sublinear variance bound for solutions of a random HamiltonJacobi equation
, 2012
"... We estimate the variance of the value function for a random optimal control problem. The value function is the solution w ǫ of a HamiltonJacobi equation with random Hamiltonian H(p,x,ω) = K(p) − V (x/ǫ,ω) in dimension d ≥ 2. It is known that homogenization occurs as ǫ → 0, but little is known abou ..."
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We estimate the variance of the value function for a random optimal control problem. The value function is the solution w ǫ of a HamiltonJacobi equation with random Hamiltonian H(p,x,ω) = K(p) − V (x/ǫ,ω) in dimension d ≥ 2. It is known that homogenization occurs as ǫ → 0, but little is known about the statistical fluctuations of w ǫ. Our main result shows that the variance of the solution w ǫ is bounded by O(ǫ/log ǫ). The proof relies on a modified Poincaré inequality of Talagrand. 1
Stochastic homogenization of nonconvex HamiltonJacobi equations in one space dimension
, 2014
"... We prove stochastic homogenization for a general class of coercive, nonconvex HamiltonJacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed. ..."
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We prove stochastic homogenization for a general class of coercive, nonconvex HamiltonJacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.
Variational formula for the timeconstant of firstpassage percolation. arXiv:1406.1108 [math
, 2014
"... Abstract. We consider firstpassage percolation with positive, stationaryergodic weights on the square lattice Zd. Let T (x) be the firstpassage time from the origin to a point x in Zd. The convergence of the scaled firstpassage time T ([nx])/n to the timeconstant as n tends to infinity can be v ..."
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Abstract. We consider firstpassage percolation with positive, stationaryergodic weights on the square lattice Zd. Let T (x) be the firstpassage time from the origin to a point x in Zd. The convergence of the scaled firstpassage time T ([nx])/n to the timeconstant as n tends to infinity can be viewed as a problem of homogenization for a discrete HamiltonJacobiBellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula for the timeconstant. As an application, we construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption), thereby computing the timeconstant. In certain situations, the iteration also produces correctors.
STOCHASTIC HOMOGENIZATION OF INTERFACES MOVING WITH CHANGING SIGN VELOCITY
"... Abstract. We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a HamiltonJacobi equation with a positively homogeneous no ..."
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Abstract. We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a HamiltonJacobi equation with a positively homogeneous noncoercive Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions and Souganidis (2009). Here we concentrate in the random media and show that the solutions of the oscillatory HamiltonJacobi equation converge in L∞weak ∗ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.
PERIODIC APPROXIMATIONS OF THE ERGODIC CONSTANTS IN THE STOCHASTIC HOMOGENIZATION OF NONLINEAR SECONDORDER (DEGENERATE) EQUATIONS
, 2013
"... Abstract. We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of HamiltonJacobi, “viscous ” HamiltonJacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We als ..."
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Abstract. We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of HamiltonJacobi, “viscous ” HamiltonJacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization. hal00851410, version 1 14 Aug 2013 1.
ASYMPTOTIC BEHAVIOR OF A NONLOCAL KPP EQUATION WITH AN ALMOST PERIODIC NONLINEARITY
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