  We extend some aspects of the Hamilton-Jacobi theory to the category of
stochastic Hamiltonian dynamical systems. More specifically, we show that the
stochastic action satisfies the Hamilton-Jacobi equation when, as in the
classical situation, it is written as a function of the configuration space
using a regular Lagrangian submanifold. Additionally, we will use a variation
of the Hamilton-Jacobi equation to characterize the generating functions of
one-parameter groups of symplectomorphisms that allow to rewrite a given
stochastic Hamiltonian system in a form whose solutions are very easy to find;
this result recovers in the stochastic context the classical solution method by
reduction to the equilibrium of a Hamiltonian system.
