  We study the large deviations principle for one dimensional, continuous,
homogeneous, strong Markov processes that do not necessarily behave locally as
a Wiener process. Any strong Markov process $X_{t}$ in $\mathbb{R}$ that is
continuous with probability one, under some minimal regularity conditions, is
governed by a generalized elliptic operator $D_{v}D_{u}$, where $v$ and $u$ are
two strictly increasing functions, $v$ is right continuous and $u$ is
continuous. In this paper, we study large deviations principle for Markov
processes whose infinitesimal generator is $\epsilon D_{v}D_{u}$ where
$0<\epsilon\ll 1$. This result generalizes the classical large deviations
results for a large class of one dimensional "classical" stochastic processes.
Moreover, we consider reaction-diffusion equations governed by a generalized
operator $D_{v}D_{u}$. We apply our results to the problem of wave front
propagation for these type of reaction-diffusion equations.
