Random, Stochastic, and Self-similar Equations

拟议项目包括三个部分。第一部分讨论了具有小随机扰动的微分方程和差分方程的李雅普诺夫指数的渐近公式，以及随机微分方程的李雅普诺夫指数的估计。第二部分研究一类随机微分方程的谱问题。 第三部分涉及自相似Dirichlet形式，扩散和随机游动的存在性和唯一性。该项目的更广泛的影响包括应用于研究随机介质中发生的自然和统计过程的长期行为。随机障碍物通道中的信号传播、等离子体中的电磁波、海洋学中的罗斯贝波、金融市场模型等都只是这类过程的几个例子。 此外，该项目有助于研究自相似物体（分形）的过程，这些过程在物理学，工程学和生物科学中有许多应用。 该项目包括各种教育和REU活动。

The proposed project consists of three parts. The first part deals with asymptotic formulas for Lyapunov exponents of differential and difference equations with small random perturbations, and estimates of the Lyapunov exponents of stochastic differential equations. The second part is devoted to the spectral problems for certain random and stochastic differential equations. The third part deals with existence and uniqueness of self-similar Dirichlet forms, diffusions, and random walks.The broader impacts of the project include applications to the study of the long term behavior of natural and statistical processes occurring in random media. Signal propagation in channels with random obstacles, electro-magnetic waves in plasma, Rossby waves in oceanography, models of financial markets are just a few of many examples of such processes. Also, the project contributes to the study of processes in self-similar objects (fractals), which have many applications in physics, engineering and biological sciences. The project includes various educational and REU activities.