A Course on Rough Paths

Abstract

We give a short overview of the scopes of both the theory of rough paths and the theory of regularity structures. The main ideas are introduced and we point out some analogies with other branches of mathematics. 1.1 Controlled differential equations Differential equations are omnipresent in modern pure and applied mathematics; many “pure” disciplines in fact originate in attempts to analyse differential equations from various application areas. Classical ordinary differential equations (ODEs) are of the form Ẏt = f(Yt, t); an important sub-class is given by controlled ODEs of the form Ẏt = f0(Yt) + f(Yt)Ẋt , (1.1) where X models the input (taking values in R, say), and Y is the output (in R, say) of some system modelled by nonlinear functions f0 and f , and by the initial state Y0. The need for a non-smooth theory arises naturally when the system is subject to white noise, which can be understood as the scaling limit as h→ 0 of the discrete evolution equation Yi+1 = Yi + hf0(Yi) + √ hf(Yi)ξi+1 , (1.2) where the (ξi) are i.i.d. standard Gaussian random variables. Based on martingale theory, Ito’s stochastic differential equations (SDEs) have provided a rigorous and extremely useful mathematical framework for all this. And yet, stability is lost in the passage to continuous time: while it is trivial to solve (1.2) for a fixed realisation of ξi(ω), after all (ξ1, . . . ξT ;Y0) 7→ Yi is surely a continuous map, the continuity of the solution as a function of the driving noise is lost in the limit. Taking Ẋ = ξ to be white noise in time (which amounts to say that X is a Brownian motion, say B), the solution map S : B 7→ Y to (1.1), known as Ito map, is a measurable map which in general lacks continuity, whatever norm one uses to