Dynamical Liouville (1805.04507v3)

Abstract: 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. One way of picturing this effect is that a naive rescaling argument suggests the above SPDE is sub-critical for all $\gamma>0$, while we don't expect solutions to exist when $\gamma > 2$. In this work, we initiate the study of this intermittent SPDE by analyzing what one might call the "classical" or "Da Prato-Debussche" phase which corresponds here to $\gamma\in[0,\gamma_{dPD}=2\sqrt{2} -\sqrt{6})$. By exploiting the positivity of the non-linearity $e^{\gamma X}$, we can push this classical threshold further and obtain this way a weaker notion of solution when $\gamma\in[\gamma_{dPD}, \gamma_{pos}=2\sqrt{2} - 2)$. Our proof requires an analysis of the Besov regularity of natural space/time Gaussian multiplicative chaos (GMC) measures. Regularity Structures of arbitrary high degree should potentially give strong solutions all the way to the same threshold $\gamma_{pos}$ and should not push this threshold further. Of independent interest, we prove along the way (using techniques from [HS16]) a stronger convergence result for approximate GMC measures which holds in Besov spaces.