We develop further ideas on how to construct low-dimensional models of stochastic dynamical systems. The aim is to derive a consistent and accurate model from the originally high-dimensional system. This is done with the support of centre manifold theory and techniques. Aspects of several previous approaches are combined and extended: adiabatic elimination has previously been used, but centre manifold techniques simplify the noise by removing memory effects, and with less algebraic effort than normal forms; analysis of associated Fokker-Plank equations replace nonlinearly generated noise processes by their long-term equivalent white noise. The ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations.

In this paper, we study the controllability of a kind of nonlinear stochastic impulsive system with infinite delay in abstract space. Sufficient conditions for the controllability are obtained via the generalization of the contraction mapping principle and with none of the compactness condition restriction on the evolution family of operators for the system. Finally, a numerical example is given to illustrate the effectiveness of our results.

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<p style="text-indent:20px;">We consider some system of complex vector fields related to the semi-classical Witten Laplacian, and establish the local subellipticity of this system basing on condition <inline-formula><tex-math id="M1">\begin{document}$ (\Psi) $\end{document}</tex-math></inline-formula>.</p>

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