We study the properties of supersymmetric models having a local Nicolai mapping. In these cases the Nicolai mapping can be interpreted as a stochastic differential equation, and hence we can use all the standard stochastic techniques to extract physical information from the theory. The corresponding Langevin equation does not describe, in general, a system approaching (asymptotically) a thermal equilibrium. We construct explicitly and nonperturbatively the Nicolai mapping for a large class of two dimensional models. In particular, this is the first non-perturbative proof of the existence of the mapping. The properties of the mapping agree with the expectations from general arguments. We show how the Nicolai mapping can be used to eliminate completely the fermions from the perturbative expansion, leaving a simpler set of diagrammatric rules involving only scalars. Finally, we argue that the present approach may be very powerful for studying finiteness properties of extended supersymmetric theories
