Abstract Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Ito formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Ito formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random “noise”, under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process.

中文翻译：

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两时间尺度随机泛函微分方程的平均原理

摘要 延迟在日常生活中无处不在、无孔不入、根深蒂固，因此有必要加以考虑。Dupire 最近开发了一个泛函 Ito 公式，它改变了随机泛函微分方程研究的格局，并鼓励人们重新考虑许多问题和应用。基于新的发展，这项工作研究了两个时间尺度的函数扩散，其中慢变过程包括路径依赖函数，而快变过程是快速变化的扩散。本文介绍中活细胞中发生的生化反应的基因表达就是这样一个鼓舞人心的例子。本文建立了混合泛函Ito公式及相应的鞅表示。然后它使用弱收敛方法开发了平均原理。通过将快变过程视为随机“噪声”，在适当的条件下，表明慢变过程弱收敛到一个随机泛函微分方程，其系数是原始慢变过程的系数的平均值。到快速变化过程的不变测度。