Regularisation-by-noise in finite and infinite dimensions

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，某些动力系统倾向于表现得更好。让我们举个例子。研究微分方程的一个主要问题是所谓的适定性，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（实际上是无穷多个）解，或者根本没有解。数学中一个显著的结果表明，对于很大一类这样的方程，只要系统受到一个随机的、足够粗糙的力的扰动，就可以恢复适定性。从数学的角度来看，这类方程非常有趣，因为它需要噪声的存在才能得到适定。此外，随着它们在应用科学中的应用越来越多，它们的重要性也超越了数学。除此之外，它们被用于工程以模拟运输-扩散现象，在金融中用于模拟股票市场，在神经科学中用于模拟相互作用的神经元。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。这个项目的目的是：1。开发能够量化粗噪声的正则化特性的技术。2. 利用这些技术来研究常微分方程和偏微分方程的适定性，这些方程表现出噪声现象的正则化。3. 研究这类方程解的数值近似。

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.