Regularisation-by-noise in finite and infinite dimensions

有限和无限维度的噪声正则化

• 批准号: 2751070
• 金额: --
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2751070
• 关键词: Regularisation; noise; finite; infinite; dimensions

Regularisation-by-noise is a branch of stochastic analysis that studies the following phenomenon: Certain dynamical systems tend to behave better when a source of randomness is present. Let us give an example. One of the main concerns in the study of differential equations is the so-called well-posedness, that is, the existence and the uniqueness of solutions. However, there are many equations that suffer from lack of well-posedness, that is, they might have multiple (in fact infinitely many) solutions or might not have a solution at all. A remarkable result in mathematics states that for a large class of those equations, well-posedness can be retrieved provided that the system is perturbed by a random (stochastic), sufficiently rough force.Equations of this type, which need the presence of the noise in order to be well-posed, are very interesting from a mathematical point of view. In addition, their importance goes beyond mathematics as they are increasingly used in the applied sciences. Among others, they are used in engineering in order to simulate transport-diffusion phenomena, in finance for modelling equity markets, and in neuroscience for modelling interacting neurons. Hence, the study of well-posedeness, qualitative properties of their solutions, and their numerical approximations is an important challenge. The aims of this project are the following: 1. Develop techniques which will allow to quantify the regularising properties of rough noises. 2. Use these techniques in order to study the well-posedness of ordinary and partial differential equations that exhibit regularisation by noise phenomena. 3. Study the numerical approximation of solutions of such equations.

噪声正则化是随机分析的一个分支，研究以下现象：当存在随机源时，某些动力系统往往表现得更好。让我们举个例子。微分方程研究的主要关注点之一是所谓的适定性，即解的存在性和唯一性。然而，有许多方程缺乏适定性，也就是说，它们可能有多个（实际上是无限多个）解，或者根本没有解。数学中的一个显着结果表明，对于一大类此类方程，只要系统受到随机（随机）、足够粗糙的力的扰动，就可以恢复适定性。此类方程需要存在噪声为了适定，从数学的角度来看非常有趣。此外，它们的重要性超出了数学范围，因为它们越来越多地用于应用科学。其中，它们在工程中用于模拟传输扩散现象，在金融中用于对股票市场进行建模，在神经科学中用于对相互作用的神经元进行建模。因此，研究其解的适定性、定性性质及其数值近似是一个重要的挑战。该项目的目标如下： 1. 开发能够量化粗糙噪声的正则化特性的技术。 2. 使用这些技术来研究通过噪声现象表现出正则化的常微分方程和偏微分方程的适定性。 3. 研究此类方程解的数值近似。