After the brilliant result of Papanicolau and Varadhan (1979) in the case of bounded stationary and ergodic environments, there has been a recent upsurge in the research of “quenched homogenization” of a symmetric diffusion process in random media. In particular, to identify the optimal conditions that a general stationary and ergodic environment must satisfy in order to obtain the convergence to a non-degenerate Brownian motion, is still an open problem. In this manuscript we show that, provided that the environment satisfies certain moment conditions, then both a quenched invariance principle and a quenched local central limit theorem hold for a diffusion formally generated by a divergence form operator. Since the coefficients are not assumed to be smooth, we shall exploit Dirichlet form theory to make sense of the diffusion associated to such operator. Both the proofs of the quenched invariance principle and of the quenched local central limit theorem rely on a priori estimates for solutions to linear partial differential equations. On one hand, with the help of the celebrated J. Moser's iteration technique, we derive a maximal inequality for solutions to degenerate elliptic PDEs which in turn gives the sublinearity of the correctors and with that the quenched invariance principle. On the other hand, relying once again on Moser's scheme, we obtain a parabolic Harnack inequality which can be used to control the oscillations of solutions to parabolic PDEs. In particular, in the diffusive limit, we are able to bound the oscillations of the transition densities of our diffusion. This successively yields the quenched local central limit theorem.
|School:||Technische Universitaet Berlin (Germany)|
|Source:||DAI-C 81/1(E), Dissertation Abstracts International|
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