Dynamical Liouville

Abstract

The aim of this paper is to analyse 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 S2 and the torus T2 in the work by David-Kupiainen-Rhodes-Vargas [14,16]. The SPDE we shall focus on has the following (simplified) form:∂tX=14πΔX−eγX+ξ, where ξ is a space-time white noise on R+×S2 or R+×T2. The main aspect which distinguishes this singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical Φ34, dynamical Sine-Gordon, generalised KPZ, 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 γ>0, while we don't expect solutions to exist when γ>2. In this work, we initiate the study of this intermittent SPDE by analysing what one might call the “classical” or “Da Prato-Debussche” phase which corresponds here to γ∈[0,γdPD=22−6). By exploiting the positivity of the non-linearity eγX, we can push this classical threshold further and obtain this way a weaker notion of solution when γ∈[γdPD,γpos=22−2). 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 γpos and should not push this threshold further unless the notion of regularity is suitably adapted to the present intermittent situation. Of independent interest, we prove along the way (using techniques from [32]) a stronger convergence result for approximate GMC measures μϵ→μ which holds in Besov spaces.