On Rough Differential Systems and Stochastic Analysis

随机微积分是概率论的一个分支，旨在研究和解释具有噪声输入的微分系统(也称为随机微分方程)。这种噪声输入最常见的例子是布朗运动，它出现在整个科学、工程和经济学的模型中。在这种背景下，所谓的粗糙路径理论是一种强大的方法，它允许人们在各种情况下定义噪声系统(远远超出布朗运动的情况)。粗糙路径理论也给出了关于随机演算的几乎确定的观点，而不是传统的方法，在本质上是高度概率的。随机分析这一术语通常包括它的随机积分和Malliavin微积分技术。后者可以被视为在路径级别定义分析的一种方式；它导致关于随机微分方程的深入和有用的结果。本研究项目旨在将粗糙路径和Malliavin演算技术相结合，以赋予并研究随机环境中模拟热传递的噪声偏微分方程组的意义。该项目还将研究由一类一般噪声驱动的微分方程的大时间行为，并了解如何通过观察典型路径来统计识别这类系统。本课题主要研究粗糙路径理论、随机偏微分方程和Malliavin演算之间的相互作用。这六个子项目的更具体的列表可以被构造如下：(1)随机偏微分方程组，具有(A)粗糙环境中的抛物型Anderson模型，(B)二维抛物型Anderson模型，以及(C)粗糙偏微分方程组的解的密度；(2)粗糙有限维系统，具有(A)粗糙微分方程的遍历性质，(B)粗糙随机微分方程的估计过程，以及(C)粗糙随机微分方程的数值格式的重整化。在所有这些项目中，首席研究员将使用随机分析方法，结合分析和粗糙路径工具，以研究一些新的但非常自然的过程类。因为在粗糙路径理论被引入之前，这些物体的定义也远不清楚，他们的研究既鼓舞人心，又具有挑战性。

Stochastic calculus is a branch of probability theory that aims to study and interpret differential systems with a noisy input (also called stochastic differential equations). The most common example of such noisy input is Brownian motion, which occurs in models throughout science, engineering, and economics. Within this context, the so-called rough paths theory is a powerful method that allows one to define noisy systems in a wide variety of situations (well beyond the Brownian motion case). The rough paths theory also gives an almost-deterministic point of view on stochastic calculus, as opposed to the traditional approach, which is highly probabilistic in essence. The term stochastic analysis usually encompasses both stochastic integration of Itô type and Malliavin calculus techniques. The latter can be viewed as a way to define an analysis at the path level; it leads to deep and useful results concerning stochastic differential equations. This research project aims to combine rough-paths and Malliavin calculus techniques in order to give a meaning to and then study noisy partial differential equations that model heat transfer in random environments. The project will also study the large time behavior of differential equations driven by a general class of noises, and see how to statistically identify this kind of system by observing a typical path. This research project focuses on interactions between rough paths theory, stochastic partial differential equations (PDE), and Malliavin calculus. A more specific list of the six subprojects can be structured as follows: (1) stochastic PDEs, with (a) parabolic Anderson model in rough environment, (b) parabolic Anderson model in dimension 2, and (c) density for solutions to rough PDEs; and (2) rough finite-dimensional systems, with (a) ergodic properties for rough differential equations, (b) estimation procedures for rough stochastic differential equations, and (c) renormalization of numerical schemes for rough stochastic differential equations. In all those projects, the principal investigator will use stochastic analysis methods combined with analysis and rough paths tools in order to study some new yet very natural classes of processes. Because even the definition of those objects was far from clear before the rough path theory was introduced, their study is both motivating and challenging.