Regularisation-by-noise in finite and infinite dimensions

噪声调节（Regulation-by-noise）是随机分析的一个分支，研究以下现象：当存在随机性时，某些动力系统倾向于表现得更好。让我们举一个例子。在微分方程的研究中，主要关注的问题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（实际上是无穷多个）解，或者可能根本没有解。数学中一个显著的结果表明，对于一个大类的方程，只要系统受到随机的、足够粗糙的力的扰动，就可以恢复适定性。这类方程需要噪声的存在才能适定性，从数学的角度来看是非常有趣的。此外，它们的重要性超越了数学，因为它们越来越多地用于应用科学。除其他外，它们被用于工程，以模拟运输扩散现象，在金融建模股票市场，并在神经科学建模相互作用的神经元。因此，研究其解的适定性、定性性质及其数值逼近是一个重要的挑战。本项目的目标如下：1.开发技术，这将允许量化粗糙噪声的正则化属性。2.使用这些技术，以研究适定性的常微分方程和偏微分方程，表现出正规化的噪声现象。3.研究这类方程解的数值逼近。

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.