Relations between different representations of stochastic adiabatic solutions

Abstract

We state the stochastic adiabatic approximation (SAA) at the formal level in a direct manner, and show that the Schöner-Haken (SH) stochastic adiabatic solution (SAS) which is defined by the partial summation of the recursion relation of SH systematic elimination procedure satisfies this formal statement. For general nonlinear stochastic differential equations (NSDE), because the conditions of the uniqueness theorem are not easily to be satisfied, we can not expect that the SH SAS is equivalent to the ‘direct’ solutions (DS). In the present study, we confine ourselves to such NSDE's in which the equation of the fast variable is linear in the fast variable. In this situation, when we deal with the NSDE's in the SAA, we get a linear SDE for the fast variable and then we conclude from the uniqueness theorem that for the same initial condition all representations of the SAS (all forms of SAS) are equivalent to each other. The procedure which transforms one form of SAS to another is proposed. This transformation is made concretely for some typical models.