Rough paths for physicists

Fri, Jan. 27th 2023, 10:00-12:30
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
Rough path theory emerged in the $1990$'s motivated by practical and conceptual questions. \\ The general context is that of controlled differential equations, for example $dU_t=f(U_t)dX_t$ where $X_t$ is a known property of a system, changing with time, and $U_t$ is an unknown whose evolution is to be computed via the equation. The emphasis is on the regularity of $X_t$ as a function of time, and its consequences on that of $U_t$, making the necessary assumptions on $f$. \\ If $X_t$ is smooth, we are on familiar grounds, but what if $X_t$, though continuous, has little regularity? In that case, it is the rule rather than the exception that Euler's discretization $U_{t_{n+1}}=U_{t_n}+f(U_{t_n})(X_{t_{n+1}}-X_{t_n})$ does not converge when the mesh goes to $0$. Approaching $X_t$ more and more closely by smooth functions $X_t^{(n)}$, and seeing if the corresponding $U_t^{(n)}$s get close to something, has its own difficulties. What is the right notion of close''? Does $\lim_{n\to \infty} U_t^{(n)}$, when it exists, depend on the approximation sequence $X_t^{(n)}$ and how? \\ Rough path theory gives an answer to these questions in a form suggestive of deep relations to physics (power counting, counterterms, etc) that I will try to explain. The short answer is that depending on the regularity of $X_t$, making unambiguous sense of the equation $dU_t=f(U_t)dX_t$ requires to supplement $X_t$ with new data which roughly (!) speaking replace some undefined integrals by a formal, yet concrete, mathematical object: a rough path structure. \\ \\ Plan: \\ 1. Motivations. The sewing lemma and Young integrals. \\ 2. Rough paths, combinatorics and regularity. A waltz with Brownian motion. \\ 3. Controlled rough paths, rough integrals and rough differential equations. \\ Rough integrals versus stochastic integrals, the It\=o stochastic area. \\ 4. Thermalization of a particle in a magnetic field. Outlook. \\ \\ Course website: https://courses.ipht.fr/?q=en/node/309

https://courses.ipht.fr/?q=en/node/309