Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows "Diffusion in a locally stationary random environment" (published in Probability Theory and Related Fields) and improves this latter work by considering possibly degenerate diffusion matrices. The geometry of the homogenized equation shows that the particle is trapped in subspace of R^d.Comment: To appear in Annales de l'institut henri poincare (link of the journal: http://www.imstat.org/aihp/

Liouville Brownian motion at criticality

In this paper, we construct the Brownian motion of Liouville Quantum Gravity with central charge $c=1$ (more precisely we restrict to the corresponding free field theory). Liouville quantum gravity with $c=1$ corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a $O(n=2)$ loop model or a $Q=4$-state Potts model embedded in a two dimensional surface in a conformal manner. Following \cite{GRV1}, we start by constructing the critical LBM from one fixed point $x\in\mathbb{R}^2$ (or $x\in\S^2$), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure $M'(dx)=-X(x)e^{2X(x)}\,dx$ (where $X$ is a Gaussian Free Field, say on $\mathbb{S}^2$). Extending this construction simultaneously to all points in $\mathbb{R}^2$ requires a fine analysis of the potential properties of the measure $M'$. This allows us to construct a strong Markov process with continuous sample paths living on the support of $M'$, namely a dense set of Hausdorff dimension $0$. We finally construct the associated Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in \cite{Rnew7,Rnew12} and also establish new capacity estimates for the critical Gaussian multiplicative chaos.Comment: 52 page

The Tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient

In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method which we believe is universal in the sense that it can be generalized to all dimensions and to all log-correlated fields. Second, in the 2d case we consider, the value of the constant in front of the inverse power is (up to explicit terms) nothing but the Liouville reflection coefficient taken at a special value. The explicit computation of the constant was performed in the recent rigorous derivation with A. Kupiainen of the DOZZ formula \cite{KRV1,KRV}; to our knowledge, it is the first time one derives rigorously an explicit value for such a constant in the tail expansion of a GMC measure. We have deliberately kept this paper short to emphasize the method so that it becomes an easily accessible toolbox for computing tails in GMC theory.Comment: The new version contains a more general statement. We also detail the relation between the Liouville reflection coefficient and the quantum spheres introduced by Duplantier-Miller-Sheffiel

Liouville Quantum Gravity on the unit disk

Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk

Complex Gaussian multiplicative chaos

In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the so-called Tachyon fields, the conformally invariant operators in critical Liouville Quantum Gravity with a c=1 central charge, and derive the original KPZ formula for these fields.Comment: 66 pages, 5 figures, contains open problems and application in 2 dimensional string theory. The new version contains the KPZ formula for the Tachyon fields and further discussions about application