618 research outputs found
Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
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
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.Comment: 57 pages. Revised version. To appear in J. Amer. Math. So
Stochastic homogenization of interfaces moving with changing sign velocity
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
Hamilton-Jacobi equation with a positively homogeneous non-coercive
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 Hamilton-Jacobi equation converge in
-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
Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments
We consider the homogenization of Hamilton-Jacobi equations and degenerate
Bellman equations in stationary, ergodic, unbounded environments. We prove
that, as the microscopic scale tends to zero, the equation averages to a
deterministic Hamilton-Jacobi equation and study some properties of the
effective Hamiltonian. We discover a connection between the effective
Hamiltonian and an eikonal-type equation in exterior domains. In particular, we
obtain a new formula for the effective Hamiltonian. To prove the results we
introduce a new strategy to obtain almost sure homogenization, completing a
program proposed by Lions and Souganidis that previously yielded homogenization
in probability. The class of problems we study is strongly motivated by
Sznitman's study of the quenched large deviations of Brownian motion
interacting with a Poissonian potential, but applies to a general class of
problems which are not amenable to probabilistic tools.Comment: 51 pages, 2 figures. We have added material and made some corrections
to our previous versio
Homogenization on arbitrary manifolds
We describe a setting for homogenization of convex hamiltonians on abelian
covers of any compact manifold. In this context we also provide a simple
variational proof of standard homogenization results.Comment: 17 pages, 1 figur
- …