Here w(t) is the «-dimensional Wiener process (Brownian motion), b(x) is a vector field, o(x) is the diffusion matrix and e ¥= 0 is a small real parameter. The cumulative effect of even very small random perturbations may be considerable after sufficiently long times, so that even if the deterministic dynamical system has an asymptotically stable equilibrium point, the trajectories of the system will leave any compact domain with probability one. The following problem was posed by Kolmogorov: determine the probability distribution of the points on the boundary where trajectories exit, at the first time of their exit from a compact domain, as well as the expected exit times. The random effect may be thought of as a slow diffusion of particles in the deterministic flow field given by b(x), and the results may differ according as particles are diffusing (a) with a flow, (b) across a flow, or (c) against a flow. Results on (a) were first obtained by Levinson [4] , and on (b) by Khasminskii [3] , both of whom used analytical techniques. Problem (c) seems to be the most difficult, and to date only partial results are available (cf. Ventsel and Freidlin [5] and Friedman [1] who used probabalistic methods). Using analytical techniques, we present a full solution of this problem for flows which are essentially gradients of a potential (as well as certain more general flows). Let f]l be a compact domain in R with a smooth boundary 3£2. Let a(j(x) = ie(o(x)a*(x))//-, be strictly positive definite in £2, b(x) = (bv b2,..., bn)9 and let u€(x) be the solution of the Dirichlet problem