Parabolic equations with a small parameter, and large deviations for diffusion processes

Nonlinear second-order parabolic equations with a small parameter at the highest derivative and coefficients depending on this parameter are considered. Under weak convergence in $L_{2,\mathrm{loc}}$ of the coefficients of the equation, uniform convergence on compacta of solutions to a generalized solution of a first-order partial differential equation is established. This result is used to justify the principle of large deviations for diffusion processes with small diffusion and coefficients that converge weakly in $L_{2,\mathrm{loc}}$.