  In contrast to the study of Langevin equations in a homogeneous environment
in the literature, the study on Langevin equations in randomly-varying
environments is relatively scarce. Almost all the existing works require random
environments to have a specific formulation that is independent of the systems.
This paper aims to consider large deviations principles (LDPs) of Langevin
equations involving a random environment that is a process taking value in a
measurable space and that is allowed to interact with the systems, without
specified formulation on the random environment. Examples and applications to
statistical physics are provided. Our formulation of the random environment
presents the main challenges and requires new approaches. Our approach stems
from the intuition of the Smoluchowski-Kramers approximation. The techniques
developed in this paper focus on the relation between the solutions of the
second-order equations and the associate first-order equations. We obtain the
desired LDPs by showing a family of processes enjoy the exponential tightness
and local LDPs with an appropriate rate function.
