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/ . The behavior is required as tends to 0 with δ small compared to . We use the theory of backward stochastic differential equations corresponding to the parabolic equation. Since δ decreases faster than , we may apply the large deviations principle with homogenized coefficients.

This paper is devoted to derive a Freidlin–Wentzell type of the large deviation principle for stochastic differential equations with general delayed generator. We improve the result of Chi Mo and Jiaowan Luo [C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 2013, 202–210].

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We present a large deviation principle for some stochastic evolution equations with jumps which depend on two small parameters, when the viscosity parameter ε tends to zero more quickly than the homogenization's one δ ε (written as a function of ε). In particular, we highlighted a large deviation principle in path-space using some classical techniques and a uniform upper bound for the characteristic function of a Feller process, in the following sense: