  The time evolution problem for non-self adjoint second order differential
operators is studied by means of the path integral formulation. Explicit
computation of the path integral via the use of certain underlying stochastic
differential equations, which naturally emerge when computing the path
integral, leads to a universal expression for the associated measure regardless
of the form of the differential operators. The discrete non-linear hierarchy
(DNLS) is then considered and the corresponding hierarchy of solvable, in
principle, SDEs is extracted. The first couple members of the hierarchy
correspond to the discrete stochastic transport and heat equations. The
discrete stochastic Burgers equation is also obtained through the analogue of
the Cole-Hopf transformation. The continuum limit is also discussed.
