The Freidlin-Wentzell LDP with rapidly growing coefficients

The Large Deviations Principle (LDP) is verified for a homogeneous diffusion process with respect to a Brownian motion $B_t$, $$ X^\eps_t=x_0+\int_0^tb(X^\eps_s)ds+ \eps\int_0^tσ(X^\eps_s)dB_s, $$ where $b(x)$ and $σ(x)$ are are locally Lipschitz functions with super linear growth. We assume that the drift is directed towards the origin and the growth rates of the drift and diffusion terms are properly balanced. Nonsingularity of $a=σσ^*(x)$ is not required.

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