Asymptotic theory of noncentered mixing stochastic differential equations

The corrected diffusion effects caused by a noncentered stochastic system are studied in this paper. A diffusion limit theorem or CLT of the system is derived with the convergence error estimate. The estimate is obtained for large t (on the interval (0,t∗), t∗ of the order of ε−1). The underlying stochastic processes of rapidly varying stochastic inputs are assumed to satisfy a strong mixing condition. The Kolmogorov–Fokker–Planck equation is derived for the transition probability density of the solution process. The result is an extension of the authorʹs previous work [J. Math. Phys. 37 (1996) 752] in that the present system is a noncentered stochastic system on the asymptotically unbounded interval. Furthermore, the solutions of the Kolmogorov–Fokker–Planck equation are represented by an explicit approximate form based upon the pseudodifferential operator theory and Wienerʹs path integral representation.