We develop a framework to discuss stability of epigenetic states as first exit problems in dynamical systems with noise. We consider in particular the stability of the lysogenic state of the lambda prophage, which is known to exhibit exceptionally large stability. The formalism defines a quantative measure of robustness of inherited states. In contrast to Kramers' well-known problem of escape from a potential well, the stability of inherited states in our formulation is not a numerically trivial problem. The most likely exit path does not go along a steepest decent of a potential -- there is no potential. Instead, such a path can be described as a zero-energy trajectory between two equilibria in an auxiliary classical mechanical system. Finding it is similar to e.g. computing heteroclinic orbits in celestial mechanics. The overall lesson of this study is that an examination of equilibria and their bifurcations with changing parameter values allow us to quantify both the stability and the robustness of particular states of a genetic control system.