In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace are identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

翻译：本文研究大规模受控随机微分方程的降维技术。所考虑的随机微分方程的漂移项包含满足单边增长条件的多项式项。此类高维设置中的非线性现象出现于例如随机反应扩散方程进行空间离散化时。我们简要讨论了解的存在性、唯一性和稳定性问题。（近似）稳定性是本文引入并研究的格拉姆矩阵新概念的基础。借助这些格拉姆矩阵，可识别主导子空间，从而建立与平衡相关的、高精度的降阶随机微分方程。我们提出了所建议模型降阶方案的代数误差准则与误差分析。最后通过将方法应用于空间离散化反应扩散方程来总结全文。