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We study large deviations in the Langevin dynamics, with damping of order$\e^{-1}$ and noise of order $1$, as $\e\downarrow 0$. The damping coefficientis assumed to be state dependent. We proceed first with a change of time andthen, we use a weak convergence approach to large deviations and theirequivalent formulation in terms of the Laplace principle, to determine the goodaction functional.  Some applications of these results to the exit problem from a domain and tothe wave front propagation for a suitable class of reaction diffusion equationsare considered.

Large Deviations for the Langevin equation with strong damping