On Rough Differential Systems and Stochastic Analysis

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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.