11,476 research outputs found
Convergence of the Abelian sandpile
The Abelian sandpile growth model is a diffusion process for configurations
of chips placed on vertices of the integer lattice , in which
sites with at least 2d chips {\em topple}, distributing 1 chip to each of their
neighbors in the lattice, until no more topplings are possible. From an initial
configuration consisting of chips placed at a single vertex, the rescaled
stable configuration seems to converge to a particular fractal pattern as . However, little has been proved about the appearance of the stable
configurations. We use PDE techniques to prove that the rescaled stable
configurations do indeed converge to a unique limit as . We
characterize the limit as the Laplacian of the solution to an elliptic obstacle
problem.Comment: 12 pages, 2 figures, acroread recommended for figure displa
Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity
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 th moment, where 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 th moment, for every , 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.Comment: Published in at http://dx.doi.org/10.1214/13-AOP833 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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