Functional integrals for generalized Fokker-Planck equation

A study is made of functional integrals representing the fundamental solution of the Fokker–Planck equation (transition probability) for the case of a degenerate diffusion matrix. Expressions are obtained for the functional integrals in configuration space in covariant and noncovariant schemes, and the corresponding Lagrangians are found (Onsager–Machlup functions). It is shown that in the noncovariant scheme the transition probability can also be expressed as a functional integral in phase space. The connection between the obtained results and the principle of least action for systems with degenerate diffusion matrix is discussed.