  We study the stochastic homogenization and obtain a random fluctuation theory
for semilinear elliptic equations with a rapidly varying random potential. To
first order, the effective potential is the average potential and the
nonlinearity is not affected by the randomness. We then study the limiting
distribution of the properly scaled homogenization error (random fluctuations)
in the space of square integrable functions, and prove that the limit is a
Gaussian distribution characterized by the homogenized solution, the Green's
function of the linearized equation around the homogenized solution, and by the
integral of the correlation function of the random potential. These results
enlarge the scope of the framework that we have developed for linear equations
to the class of semilinear equations.
