Stochastic evolution equations within the context of both the Hamiltonian and Lagrangian formalisms

It is proved that it is possible to obtain continuum deterministic and stochastic evolution equations from a set of discrete stochastic rules after an average over realizations and over near neighbors or coarse graining on the dynamical variables, respectively. Examples are given that allow us to find the Hamilton evolution equation for the dynamical variables and the Euler evolution equation for the Lagrangian of the system with additive noises.