On Some Applicable Versions of Abstract Large Deviations Theorems

Abstract

Two algorithms for calculating rate functions for a family of measures $\{\mu_\varepsilon\}$ on a $B$-space $X$ are considered. The first one is a relaxed version of the Fenchel transform type theorem for convex rate functions. The second gives conditions under which $\{\mu_\varepsilon\}$ can be replaced by a more convenient family $\{\mu^x_\varepsilon\}$ near admissible points $x \in X$ such that rate functions for both families coincide near $x$. As an example, we apply both techniques to investigate large deviation properties of some reaction-diffusion equations with quick random noise.