Theoretical and Numerical Studies of Nonlocal Equations Derived from Stochastic Differential Equations with Levy Noises

随机效应在科学和工程中的复杂系统中普遍存在。虽然随机机制可能看起来非常小或非常快，但它们对系统演化的长期影响可能是微妙的甚至是深远的，这已经在例如随机分叉、随机共振和噪声诱导的模式形成中观察到。研究小组将通过开发在计算机上模拟的数值方法来研究不确定条件下的复杂系统，并回答有关系统平均量的基本问题。调查人员将产生以下更广泛的影响：(1)两名研究生(包括一名女性代表不足的少数民族)将接受教育和培训；(2)他们将继续在STEM招收和培养代表不足的学生。此外，由此产生的计算工具在从生物学到地球物理学的各个领域都有广泛的应用。所提出的项目所产生的软件将被公开使用。对不确定条件下的复杂系统进行数学建模通常会产生随机微分方程(SDE)。SDE中出现的波动通常是非高斯的(例如，Levy运动)，而不是高斯的(例如，布朗运动)。与具有高斯噪声的系统相比，量化非高斯列维涨落的影响的了解要少得多。研究人员将开发收敛和高效的数值技术来研究确定性的宏观量，这些技术可以帮助理解带有Levy噪声的SDE的动力学，特别是平均退出时间、逃逸概率和概率密度函数。他们还将回答有关这些非局部方程解的适定性和正则性的理论问题。在先前开发的一维情况下的方法的基础上，该团队将专注于二维系统。拟议的项目不仅将提供广泛适用的计算技术来求解具有奇异积分的积分-微分方程组，而且还将在理论上解决与解有关的中心问题。这一理论将有助于对动力学系统中非高斯列维涨落的影响进行量化的基本问题提供洞察。

Stochastic effects are ubiquitous in complex systems in science and engineering. Although random mechanisms may appear to be very small or very fast, their long time impact on the system evolution may be delicate or even profound, which has been observed in, for example, stochastic bifurcation, stochastic resonance and noise-induced pattern formation. The research team will study the complex systems under uncertainty by developing numerical methods to be simulated on computers and answer fundamental questions about the average quantities of the systems. The investigators will deliver the following broader impact outcomes: (1) Two graduate students (including one female underrepresented minority) will receive education and training and (2) they will continue to recruit and nurture underrepresented students in STEM. Additionally, the resulting computational tools have broad applications in areas ranging from biology to geophysics. The software resulting from the proposed project will be made publicly available.Mathematical modeling of complex systems under uncertainty often leads to stochastic differential equations (SDEs). Fluctuations appeared in the SDEs are often non-Gaussian (e.g., Levy motions) rather than Gaussian (e.g., Brownian motion). Compared with systems with Gaussian noises, quantifying the impacts of non-Gaussian Levy fluctuations are much less understood. The researchers will develop convergent and efficient numerical techniques for investigating the deterministic macroscopic quantities that can help understand the dynamics of SDEs with Levy noises, in particular the mean exit time, escape probability, and probability density function. They will also answer theoretical questions with regard to the well-posedness and the regularity of the solutions to these nonlocal equations. Building upon previously developed methods in the one-dimensional case, the team will focus its effort on two-dimensional systems. The proposed project will provide not only broadly applicable computational techniques to solve integro-differential equations with singular integrands, but also theoretically address the central questions pertaining to the solutions. The theory will help to provide insight into fundamental issues in quantifying the impacts of non-Gaussian Levy fluctuations in dynamical systems.