  The Large Deviations Principle (LDP) is verified for a homogeneous diffusion
process with respect to a Brownian motion $B_t$, $$
X^\eps_t=x_0+\int_0^tb(X^\eps_s)ds+ \eps\int_0^t\sigma(X^\eps_s)dB_s, $$ where
$b(x)$ and $\sigma(x)$ 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
$a=\sigma\sigma^*(x)$ is not required.
