On the Contraction of Fast Driving Variables from Stochastic Processes

概要

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A class of Langevin equations where slow variables x are driven by fast variables y (generally, nonlinear with regard to x) is treated to obtain the explicit from of the contraction (an asymptotic x process equivalent to the original process in a large time scale) by a limit theorem of Papanicolaou, Strook and Varadhan, with which the recent work of the same problem by Morita, Mori and Mashiyama is reexamined. It is shown that a non-uniqueness of the contracted drift exists that reflects the lack of a proper prescription either of assigning the convergence factor ε in the limit theorem or of choosing the projector in the projector elimination method. In fact, the so-called "local equilibrium assumption" is not always a relevant choice, as can be seen in the simple example of the Ornstein-Uhlenbeck process in a binding force filed. A general prescription is given to unify two different situations by decomposing the y-drift into two parts ; a part of binding the x-subsystem and a part of producing its macroscopic, sourceless flow. This yields a more general result of the contraction formula. It is argued that, apart from this distinction, the contracted result is independent of the choice of the projector in case that one goes to higher order terms of ε.
理論物理学刊行会の論文
1982-01-25