Dynamical Liouville

Understand

The aim of this paper is to analyze an SPDE which arises naturally in the context of Liouville quantum gravity.

• This SPDE shares some common features with the so-called Sine-Gordon equation and is built to preserve the Liouville measure which has been constructed recently on the two-dimensional sphere $S^2$ and the torus $T^2$ in the work by David-Kupiainen-Rhodes-Vargas.
• The SPDE we shall focus on has the following (simplified) form: \[ p_t X = \frac 1 {4\pi} \Delta X - e^{\gamma X} + \xi\,, \] where $\xi$ is a space-time white noise on $R_+ \times S^2$ or $R_+ \times T^2$.
• The main aspect which distinguishes this singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical $\Phi^4_3$, dynamical Sine-Gordon, etc.) is the presence of intermittency.