标题： 稀释自催化化学网络的最优多时间尺度估计。 (1) 引言和$σ^*$-主导情况 显示英文标题

标题： Optimal multi-time-scale estimates for diluted autocatalytic chemical networks. (1) Introduction and $σ^*$-dominant case

摘要： 自催化化学网络是动力系统，其在零点附近的线性化具有正的李雅普诺夫指数；该指数给出了稀释区域中系统的增长速率，即对于近零浓度的情况。 在动力学极限中，动力学的生成元是一个佩龙-弗罗贝尼乌斯矩阵，这表明可以使用马尔可夫链技术来获得长时间的渐近行为。 这一系列研究介绍了一种新的、通用的方法，该方法提供了关于此类渐近行为的精确定量信息，基于对李雅普诺夫特征值和特征向量的估计。 该算法受到量子场论中威尔逊重正化群方法的启发，基于从最快到最慢的动能尺度的向下递归。 估计形式为动能速率的简单有理函数。 它们在尺度分离假设下是准确的，该假设松散地表示动能速率跨越了许多数量级。 我们在这里提供该方法的一般动机和简介，展示一些简单的例子，并推导出若干初步结果，特别是对所谓$\sigma^*$-主导图的一个子类的李雅普诺夫数据的估计。 显示英文摘要

摘要： Autocatalytic chemical networks are dynamical systems whose linearization around zero has a positive Lyapunov exponent; this exponent gives the growth rate of the system in the diluted regime, i.e. for near-zero concentrations. The generator of the dynamics in the kinetic limit is then a Perron-Frobenius matrix, suggesting the use of Markov chain techniques to get long-time asymptotics. This series of works introduces a new, general procedure providing precise quantitative information about such asymptotics, based on estimates for the Lyapunov eigenvalue and eigenvector. The algorithm, inspired from Wilson's renormalization group method in quantum field theory, is based on a downward recursion on kinetic scales, starting from the fastest, and terminating with the slowest rates. Estimates take on the form of simple rational functions of kinetic rates. They are accurate under a separation of scales hypothesis, loosely stating that kinetic rates span many orders of magnitude. We provide here a brief general motivation and introduction to the method, present some simple examples, and derive a number of preliminary results, in particular the estimation of Lyapunov data for a subclass of so-called $\sigma^*$-dominant graphs.