Abstract
We consider the theory of correctors to homogenization in stationarytransport equations with rapidly oscillating, random coefficients.Let ε << 1 be the ratio of the correlation length in the randommedium to the overall distance of propagation. As ε $ \downarrow0$, we show that the heterogeneous transport solution iswell-approximated by a homogeneous transport solution. We then showthat the rescaled corrector converges in (probability) distributionand weakly in the space and velocity variables, to a Gaussianprocess as an application of a central limit result. The latterresult requires strong assumptions on the statistical structure ofrandomness and is proved for random processes constructed bymeans of a Poisson point process.
Highlights
Partial differential equations with rapidly varying coefficients arise naturally in many important applications, such as e.g. composite material sciences, nuclear sciences, porous media equations, and Earth science as in e.g. climate modeling
We consider the stationary case here the results extend to the evolution equation as well
We develop a theory for the random corrector
Summary
Partial differential equations with rapidly varying coefficients arise naturally in many important applications, such as e.g. composite material sciences, nuclear sciences, porous media equations, and Earth science as in e.g. climate modeling. Homogenization theory for transport equations is well understood in fairly arbitrary ergodic random media, see e.g. Whereas the results should hold for more general processes, it is clear that much more severe restrictions than mere ergodicity as in [22] must be imposed on the random structure in order to obtain a full characterization of the limiting behavior of the corrector. This is the case for elliptic equations as may be seen in e.g.
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