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 Itô stochastic area.
4. Thermalization of a particle in a magnetic field. Outlook.