1.5
Let 
be a probability space and suppose there exist invariant ergodic
translation operators 
, which act on . Assume
is a measurable function from
to the space of 
symmetric matrices. Assume
further that 
is uniformly elliptic. That is there exist constants
, such that
This paper is concerned with solutions of the random elliptic partial difference equation,
where 
is a 
function of compact support.
Here 
denotes discrete gradient on 
and
its adjoint.
According to a theorem of Papanicolau-Varadhan (``Boundary value problems
with rapidly oscillating random coefficients", Volume 2 of Coll. Math.
Soc. Janos Bolya 27, Random Fields, North Holland 1981),
converges in mean square as 
to
a function 
which satisfies a constant coefficient
elliptic equation on 
. Evidently u(x) is a smooth function. In
contrast, one cannot expect a typical solution of the
random equation to be smooth i.e. discrete derivatives of
bounded independent of 
as
, but one might expect derivatives of the
expectation of to be bounded independent of
. This is proved in the paper.
The paper also studies the rate of convergence of the variance of
to zero. A rate is obtained provided the variables
, are assumed to be independent or weakly
correlated. The rate depends only on the ratio 
. The proof
requires the use of Legendre polynomials, in particular the fact that
. This result
complements previous work of Spencer and Naddaf (``Estimates on the variance
of some homogenisation problems", preprint, 1998). In their work
the probability space must be a Euclidean field theory which
satisfies the Brascamp-Lieb inequality.