Application of homogenization and large deviations to a parabolic semilinear equation

We study the behavior of the solution of a partial differential equation with a linear parabolic operator with non-constant coefficients varying over length scale δ and nonlinear reaction term of scale 1/ϵ. The behavior is required as ϵ tends to 0 with δ small compared to ϵ. We use the theory of backward stochastic differential equations corresponding to the parabolic equation. Since δ decreases faster than ϵ, we may apply the large deviations principle with homogenized coefficients.