论文标题
用于计算由随机微分方程控制的系统中功能的期望值的组合。
Combinatorics for calculating expectation values of functions in systems with evolution governed by stochastic differential equations
论文作者
论文摘要
随机微分方程广泛用于各个领域。特别是，在某些模型（例如人口模型和布朗动量过程）中已经证明了二元关系的有用性。在这项研究中，进行了基于组合学的讨论并应用于计算函数的期望值，在该系统中，进化由随机微分方程控制的系统。从随机过程的二元性理论开始，对时间顺序运算符的解释和使用进行了一些修改，自然会导致有关组合学的讨论。为了进行演示，Ornstein-Uhlenbeck过程的第一时刻和第二瞬间是从有关组合学的讨论中重新衍生的。此外，提出了两种用于实际应用的数值方法。一种方法是基于常规的指数膨胀和幻象近似。另一个使用时间进化运算符的分解，以及Aitken系列加速方法的应用。两种方法都产生合理的近似值。尤其是，分解和AITKEN加速度显示出令人满意的结果。这些发现将提供一种新的方式来直接和直接地计算期望，而无需使用时间浪费。
Stochastic differential equations are widely used in various fields; in particular, the usefulness of duality relations has been demonstrated in some models such as population models and Brownian momentum processes. In this study, a discussion based on combinatorics is made and applied to calculate the expectation values of functions in systems in which evolution is governed by stochastic differential equations. Starting with the duality theory of stochastic processes, some modifications to the interpretation and usage of time-ordering operators naturally lead to discussions on combinatorics. For demonstration, the first and second moments of the Ornstein-Uhlenbeck process are re-derived from the discussion on combinatorics. Furthermore, two numerical methods for practical applications are proposed. One method is based on a conventional exponential expansion and the Pade approximation. The other uses a resolvent of a time-evolution operator, along with the application of the Aitken series acceleration method. Both methods yield reasonable approximations. Particularly, the resolvent and Aitken acceleration show satisfactory results. These findings will provide a new way of calculating expectations numerically and directly without using time-discretization.
