  Stochastic mechanics---the study of classical stochastic systems governed by
things like master equations and Fokker-Planck equations---exhibits striking
mathematical parallels to quantum mechanics. In this article, we make those
parallels more transparent by presenting a quantum mechanics-like formalism for
deriving a path integral description of systems described by stochastic
differential equations. Our formalism expediently recovers the usual path
integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup
forms) and is flexible enough to account for different variable domains (e.g.
real line versus compact interval), stochastic interpretations, arbitrary
numbers of variables, explicit time-dependence, dimensionful control
parameters, and more. We discuss the implications of our formalism for
stochastic biology.
