Iterated integrals and controlled ODEs

Abstract: The purpose of this non-technical talk is to discuss the role that iterated integrals (an instance of the so-called mapping space signature) play in the theory of ordinary differential equations (ODEs) of the form

\[ \dot{Y}(t) = f(Y(t))\dot{X}(t)\]

where the path $X$ models the input and the path $Y$ models the output of a physical system whose dynamics is governed by a (non-linear function) $f$. When $X$ is sufficiently regular (e.g.\ Lipschitz) function, the solution to the ODE can be, roughly speaking, found as a limit of its iterated integrals. Such regularity is, however, typically not obtained if $X$ is a sample path of a stochastic process (e.g.\ the Wiener process) and $\dot X$ models the random noise (e.g.\ the white noise). In the talk, we will discuss why low regularity is a problem, what can be done, and why it is important to understand the algebraical and analytical properties of the iterated integrals (and, more generally, the mapping space signature).

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.