Complex paths for real stochastic processes

The calculation of the decay rate of a metastable state in the path-integral formulation of stochastic processes is revisited. Previous derivations of this rate were achieved at the cost of a step that is difficult to justify mathematically. We show that this difficulty can be resolved by working with a complex extremal solution that arises naturally in the Itô formulation of the path integral. To make the analysis as transparent as possible, we choose a simple potential for which the extremal solution can be written in terms of elementary functions. The mechanism identified here, however, is not restricted to this example and holds more generally.