摘 要：随机游走模型作为概率论中的重要分支，在物理、化学、生物、经济等领域有着广泛应用。本文旨在深入研究随机游走模型，基于经典理论框架，引入时间相关性和空间异质性因素，构建了更为复杂的高维随机游走模型。通过引入分数阶微积分工具，对传统随机游走模型进行改进，解决了现有模型在描述复杂系统时的局限性。研究采用蒙特卡罗模拟方法与解析推导相结合的方式，对不同参数条件下的随机游走特性进行了系统分析。结果表明，改进后的模型能够更准确地刻画实际系统的扩散行为，特别是在非均匀介质和长程相关性场景中表现出色。该研究不仅丰富了随机过程理论体系，还为跨学科应用提供了新的理论依据，特别是为理解复杂系统的时空演化机制提供了有效工具。通过对随机游走模型的创新性拓展，本研究为后续相关领域的理论发展和实际应用奠定了坚实基础。
关键词：随机游走模型；分数阶微积分；时间相关性
Abstract：The random walk model, as an important branch of probability theory, has extensive applications in fields such as physics, chemistry, biology, and economics. This study aims to delve into the random walk model by introducing time-correlation and spatial heterogeneity factors within the fr amework of classical theory, thereby constructing a more complex high-dimensional random walk model. By incorporating fractional calculus tools, this research improves upon traditional random walk models, addressing their limitations in describing complex systems. The study employs a combination of Monte Carlo simulation methods and analytical derivations to systematically analyze the characteristics of random walks under various parameter conditions. Results indicate that the improved model can more accurately depict the diffusion behavior of actual systems, particularly excelling in non-uniform media and long-range correlation scenarios. This research not only enriches the theoretical fr amework of stochastic processes but also provides new theoretical foundations for interdisciplinary applications, especially in understanding the spatiotemporal evolution mechanisms of complex systems. Through innovative extensions of the random walk model, this study lays a solid foundation for future theoretical development and practical applications in related fields.
引言 1
一、随机游走的基本概念与理论 1
(一)随机游走的定义与发展历程 1
(二)一维随机游走的概率特性 2
(三)多维随机游走的扩展研究 2
二、随机游走的概率分布分析 3
(一)步长分布的影响因素 3
(二)游走位置的概率密度函数 3
(三)极限定理在随机游走中的应用 4
三、随机游走模型的应用场景 4
(一)物理系统中的随机游走现象 5
(二)生物过程中的随机运动模型 5
(三)金融市场中的随机游走假设 6
四、随机游走模型的现代发展 6
(一)分数阶随机游走模型 6
(二)相互作用粒子系统的游走 7
(三)随机环境下的游走特性 7
结论 8
参考文献 9
致谢 9
关键词：随机游走模型；分数阶微积分；时间相关性
Abstract：The random walk model, as an important branch of probability theory, has extensive applications in fields such as physics, chemistry, biology, and economics. This study aims to delve into the random walk model by introducing time-correlation and spatial heterogeneity factors within the fr amework of classical theory, thereby constructing a more complex high-dimensional random walk model. By incorporating fractional calculus tools, this research improves upon traditional random walk models, addressing their limitations in describing complex systems. The study employs a combination of Monte Carlo simulation methods and analytical derivations to systematically analyze the characteristics of random walks under various parameter conditions. Results indicate that the improved model can more accurately depict the diffusion behavior of actual systems, particularly excelling in non-uniform media and long-range correlation scenarios. This research not only enriches the theoretical fr amework of stochastic processes but also provides new theoretical foundations for interdisciplinary applications, especially in understanding the spatiotemporal evolution mechanisms of complex systems. Through innovative extensions of the random walk model, this study lays a solid foundation for future theoretical development and practical applications in related fields.
Keywords: Random Walk Model；Fractional Calculus；Time Correlation
引言 1
一、随机游走的基本概念与理论 1
(一)随机游走的定义与发展历程 1
(二)一维随机游走的概率特性 2
(三)多维随机游走的扩展研究 2
二、随机游走的概率分布分析 3
(一)步长分布的影响因素 3
(二)游走位置的概率密度函数 3
(三)极限定理在随机游走中的应用 4
三、随机游走模型的应用场景 4
(一)物理系统中的随机游走现象 5
(二)生物过程中的随机运动模型 5
(三)金融市场中的随机游走假设 6
四、随机游走模型的现代发展 6
(一)分数阶随机游走模型 6
(二)相互作用粒子系统的游走 7
(三)随机环境下的游走特性 7
结论 8
参考文献 9
致谢 9
