SIAM Review, Volume 63, Issue 1, Page 1-1, January 2021.
The use of probabilistic ideas by applied mathematicians has seen a continued increase in recent decades. Probability now appears frequently at the modeling stage. There is widespread interest in investigating the effects of noise and uncertainty. Probabilistic algorithms are routinely applied with much success to the solution of deterministic problems (think of Monte Carlo quadrature or stochastic gradient descent). Markov chains give a simple, widely used tool to describe systems that evolve randomly. They are also a very popular topic in applied mathematics courses. At times \(0\), \(1\), \(2\), łdots, a Markov chain jumps from its current state \(x\) to a randomly chosen new state \(y\); the set of possible states is discrete. The requirement that the jumping times be uniformly spaced is a clear limitation in many situations, and this shortcoming is avoided by considering continuous-time Markov chains. In the continuous-time setting, if the chain has reached state \(x\) at time \(t_i\), it will jump to the randomly chosen \(y\) at time \(t_i+1=t_i+\tau_i\), where the waiting time \(\tau_i>0\) is itself random. Markov chains in continuous time are featured in many applications, including chemistry, ecology, and epidemiology. In a chemical system containing \(n\) species \(S_1\), łdots, \(S_n\), each state corresponds to vector \((c_1\), łdots, \( c_n)\) where \(c_j\) is the number of molecules of species \(S_j\). The species may take part in a number of chemical reactions, let us say \(S_1+2S_2 \rightarrow S_3\), \(2S_1+3S_3 \rightarrow S_4+2S_6\), and so on. At random times, one or another of the \(m\) possible reactions will take place and the state will change. Such a stochastic, molecular approach typically provides a more accurate description of the system than deterministic treatments where the concentrations of the different species are regarded as continuous variables that evolve according to a set of differential equations. This is particularly true in systems where, for some species, the number \(c_j\) is low, as is the case in many biological reactions. The following Survey and Review paper, “Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations,” has been written by Juan Kuntz, Philipp Thomas, Guy-Bart Stan, and Mauricio Barahona. Section 2 provides a reader-friendly introduction to continuous-time Markov chains and their stationary distributions; these are important because they determine the long-time behavior of the chain. A salient future is that the authors present, in an accessible way, results that do not assume the chain to be irreducible (roughly speaking, irreducibility means that the chain is not the juxtaposition of two or more smaller chains that do not talk to each other; irreducibility simplifies the math, but is not a reasonable hypothesis in some applications). After that, the authors concentrate on how to compute invariant distributions. The paper contains numerical results, an extensive bibliography, and a detailed discussion of open problems.

中文翻译：

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调查和审查

SIAM 评论，第 63 卷，第 1 期，第 1-1 页，2021 年 1 月。
近几十年来，应用数学家对概率思想的使用持续增加。概率现在经常出现在建模阶段。研究噪声和不确定性的影响引起了广泛的兴趣。概率算法通常成功地应用于确定性问题的解决方案（想想蒙特卡罗正交或随机梯度下降）。马尔可夫链提供了一个简单的、广泛使用的工具来描述随机演化的系统。它们也是应用数学课程中非常受欢迎的话题。有时\(0\)、\(1\)、\(2\)、\dots，马尔可夫链从其当前状态\(x\)跳到随机选择的新状态\(y\)；可能的状态集是离散的。跳跃时间均匀间隔的要求在许多情况下是一个明显的限制，并且通过考虑连续时间马尔可夫链来避免这个缺点。在连续时间设置中，如果链在时间\(t_i\)达到状态\(x\)，它将在时间\(t_i+1=t_i+\tau_i\)跳转到随机选择的\(y\) )，其中等待时间 \(\tau_i>0\) 本身是随机的。连续时间的马尔可夫链在许多应用中都有特色，包括化学、生态学和流行病学。在包含 \(n\) 种 \(S_1\), łdots, \(S_n\) 的化学系统中，每个状态对应于向量 \((c_1\), łdots, \(c_n)\) 其中 \(c_j\ ) 是物种\(S_j\) 的分子数。物种可能参与许多化学反应，让我们说 \(S_1+2S_2 \rightarrow S_3\)、\(2S_1+3S_3 \rightarrow S_4+2S_6\) 等等。在随机时间，\(m\) 个可能的反应中的一个或另一个将发生并且状态将改变。这种随机的分子方法通常比确定性处理提供更准确的系统描述，其中不同物种的浓度被视为根据一组微分方程演变的连续变量。在某些物种的数量 \(c_j\) 很低的系统中尤其如此，就像许多生物反应的情况一样。以下调查和评论论文“连续时间马尔可夫链的平稳分布：理论和基于截断的近似值回顾”由 Juan Kuntz、Philipp Thomas、Guy-Bart Stan 和 Mauricio Barahona 撰写。第 2 节对连续时间马尔可夫链及其平稳分布进行了对读者友好的介绍；这些很重要，因为它们决定了链条的长期行为。一个显着的未来是，作者以一种易于理解的方式呈现出不假设链不可约的结果（粗略地说，不可约意味着该链不是两个或更多彼此不交谈的较小链的并置; 不可约性简化了数学，但在某些应用中不是合理的假设）。之后，作者专注于如何计算不变分布。该论文包含数值结果、广泛的参考书目以及对开放问题的详细讨论。但在某些应用中不是合理的假设）。之后，作者专注于如何计算不变分布。该论文包含数值结果、广泛的参考书目以及对开放问题的详细讨论。但在某些应用中不是合理的假设）。之后，作者专注于如何计算不变分布。该论文包含数值结果、广泛的参考书目以及对开放问题的详细讨论。