We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in . If M is a non degenerate multifractal measure with associated metric ρ(x, y) = M ([x, y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dim H of a measurable set K and the… (More)
In this article, we review the theory of Gaussian multiplica-tive chaos initially introduced by Kahane's seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2d-Liouville quantum gravity or… (More)
This paper deals with homogenization of diffusion processes in a locally stationary random environment. Roughly speaking, such an environment possesses two evolution scales: both a fast microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims at giving a macro-scopic approximation that takes into account the microscopic… (More)
This paper is concerned with the construction of atomic Gaussian multi-plicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ 2 beyond the transition phase (i.e. γ 2 > 2d) and check the duality relation with sub-critical Gaussian… (More)
We construct and study space homogeneous and isotropic random measures (MMRM) which generalize the so-called MRM measures constructed in . Our measures satisfy an exact scale invariance equation (see equation (1) below) and are therefore natural models in dimension 3 for the dissipation measure in a turbulent flow.
We study a diffusion with random, time dependent coefficients. Moreover, the diffusion coefficient is allowed to degenerate. We prove the invariance principle when this diffusion is supposed to be controlled by another one with time independent coefficients.
In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the… (More)
We investigate stochastic homogenization for some degenerate quasilinear pa-rabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium. Assuming that this reference diffusion operator is… (More)
Gaussian Multiplicative Chaos is a way to produce a measure on R d (or subdo-main of R d) of the form e γX(x) dx, where X is a log-correlated Gaussian field and γ ∈ [0, √ 2d) is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between −∞ and ∞ and is not a function in the usual sense. This procedure yields the… (More)