Abstract: The extent to which a stochastic process admits an associated rough path is, arguably, one of its fundamental aspects. In particular, this enables the subsequent use of rough analysis, a purely deterministic theory, that has proven to be the most powerful tool in stochastic analysis. This has been carried out in great generality for classes of Gaussian processes, semimartingales and Markov processes, e.g. [F-Victoir, CUP 2010].
After a quick survey what can, and cannot, be done, I will talk about (a very early) project with Y. Yuan (Berlin) and S. Andres (Manchester) where we study a canonical rough path associated to Liouville Brownian motion. This process was introduced independently by Berstycki and Garban--Rhodes--Vargas with the aim to "introduce a whole set of tools of stochastic analysis in Liouville quantum gravity (LQG), which will be hopefully useful in analyzing the geometry of LQG". Our investigation can be seen as a first attempt to inject ideas from rough path theory into LGQ.