I have a system described by the Langevin equation da/dt = - dF/da* + r where a are complex amplitudes of electromagnetic modes (and r is the white noise). How if F was real, it would be the Hamiltonian of the system, but in my case (and in general), F is complex (because the a are complex themselves). So F cannot be an Hamiltonian. How can I obtain a Hamiltonian formulation of this problem? P.S.: my first idea was to write to separate equation for the real and imaginary parts of a. But in this case I obtain two Hamiltonian, both depending on both Re[a] and Im[a], so which is the Hamiltonian of the system? If e.g. I want to use a Gibbs measure for a, what I must use?