Metastability for parabolic equations with drift: part 1 H. IshiiP. Souganidis 35B4035K2037H99parabolic equationasymptotic behaviormetastabilitystochastic perturbation We study the exponentially long time behavior of solutions to linear uniformly parabolic equations which are small perturbations of transport equations with vector fields having a globally stable (attractive) equilibrium in the domain. The result is 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. Problems of this type were considered by Freidlin and Wentzell and Freidlin and Koralov, using probabilistic arguments related to the theory of large deviations. Our approach, which is self-contained, relies entirely on pde arguments, and is flexible to the extent that allows us to study a class of semilinear equations of similar structure. This note also prepares the ground for the forthcoming Part II of this work where we consider general quasilinear problems. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5559 10.1512/iumj.2015.64.5559 en Indiana Univ. Math. J. 64 (2015) 875 - 913 state-of-the-art mathematics http://iumj.org/access/