  We provide a self-contained analysis, based entirely on pde methods, of the
exponentially long time behavior of solutions to linear uniformly parabolic
equations which are small perturbations of a transport equation with vector
field having a globally stable point. We show that the solutions converge to a
constant, which is either the initial value at the stable point or the boundary
value at the minimum of the associated quasi-potential. This work extends
previous results of Freidlin and Wentzell and Freidlin and Koralov and applies
also to semilinear elliptic pde.
