arXiv subjects

André L. P. Considera

Publications and source records attributed to André L. P. Considera.

3 recordsLinked to original sources

arXiv2026

Renormalization Group on Wiener Space: Spectral Theory and Universality

We develop a rigorous renormalization group (RG) formalism on Wiener space. We introduce an RG operator $\mathcal R$ acting on probability measures over continuous paths, whose fixed point is the Wiener measure. We linearize $\mathcal{R}$ around the Brownian fixed point and analyze the spectral structure of the resulting operator $\mathscr{L}$. Its eigenvectors are too singular to be realized as honest measures on path space, and we therefore develop a generalized spectral theory within the framework of white noise analysis (Hida calculus). The eigenvectors of $\mathscr{L}$ are realized as Hida distributions, satisfying $\mathscr{L} U=λU$ in the weak sense. This analysis yields structural results such as a spectral gap and a diagonal-concentration property of the eigenvectors. Moreover, we identify a family $\{\mathfrak{U}_n\}_{n \geq 0}$ of eigenvectors with eigenvalues $λ_n = 2^{1-n/2}$, consisting of Wick polynomials of white noise formally expressed as $\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\, dt$, and rigorously constructed as Hida distributions. We then use this spectral structure to uncover a finer, second layer of universality in the Donsker invariance principle: not only is the convergence of random walks towards the Brownian scaling limit universal, but the entire hierarchy of leading corrections to the Brownian limit is universal as well, governed by the eigenpairs $(λ_n, \mathfrak{U}_n)$. We also show, at a formal level, that the framework developed here extends to Gibbs-type perturbations of the Brownian fixed point, in the spirit of self-interacting quantum field theories. In particular, we recover the standard irrelevant/marginal/relevant classification, usually obtained in the physics literature by power-counting, solely from the spectral analysis of $\mathscr{L}$.

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arXiv2026

Simulating passive scalar advection in rough Kraichnan flows

We present a systematic Eulerian study of passive scalar advection in rough two-dimensional Kraichnan flows, covering the full range of velocity roughness exponent $h\in(0,1)$. The advection--diffusion equation is integrated directly using a pseudo-spectral method at resolutions up to $2048^2$ grid points, with a frozen-noise Runge--Kutta scheme consistent with the white-in-time construction of the carrier flow. The simulations recover the duality between advected scalar and advecting flow---the smoother the carrier, the rougher the scalar---together with the predicted scaling laws for the second-order statistics. Once the scaling range is properly identified, the fourth-order flatness anomaly is measured across the whole range of $h$, in agreement with the Lagrangian estimates of Frisch \textit{et al.} (1999) and with the perturbative predictions of Bernard \textit{et al.} (1998) and Pumir \textit{et al.} (1997). Benefiting from the fact that our Eulerian simulations give direct access to the full scalar field, we also examine the probability density functions of scalar increments and the corresponding higher-order statistics, which show systematic departures from Gaussianity and from log-normality, with the strongest deviations manifesting for intermediate values of $h$. A central outcome of this work is a systematic account of how the simulation parameters, in particular the molecular diffusivity, must be adjusted with $h$, providing practical guidelines for reliable simulations; we further show that the residual deviations from the theoretical scaling laws are quantitatively accounted for by the finite spectral representation of the carrier flow. Our analysis also serve as a numerical baseline for simulating more realistic extensions of the Kraichnan model, where the carrier flow is coupled with a Gaussian multiplicative chaos and theoretical developments are limited.

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arXiv2025

Transport in multifractal Kraichnan flows: from turbulence to Liouville quantum gravity

We investigate the behavior of fluid trajectories in a multifractal extension of the Kraichnan model of turbulent advection. The model couples a one-dimensional, Gaussian, white-in-time random flow to a frozen-in-time Gaussian multiplicative chaos (GMC). The resulting velocity field features an interplay between the roughness exponent $ξ\in(0,2]$, controlling the correlation decay for the Gaussian component, and the intermittency parameter $γ\in [0,\sqrt {2}/2)$, prescribing the deviations from self-similarity. Recent numerical work by the authors suggests that such coupling induce a smoothing-by-intermittency effect, and the purpose here is to address this phenomenon theoretically. Using the theory of 1D Feller Markov processes, we characterize the phases of the two-particle separation process upon varying $ξ$ and $γ$, extending to a multifractal setting the stochastic/deterministic and colliding/non-colliding transitions known in the monofractal Kraichnan case. Our analysis distinguishes between two settings: quenched or annealed. In the quenched setting, the GMC realization is prescribed, and we show that the phases are governed by the most probable Hölder exponent of the multifractal velocity field. In the annealed setting, the GMC is averaged over, leading to an additional smoothing effect. Moreover, we show that the separation process exhibits structural analogies with multiplicative one-dimensional versions of the Liouville Brownian motion (LBM)--a diffusion process evolving in a random GMC landscape, originally introduced in the context of Liouville quantum gravity. In particular, both the quenched and annealed phase transitions are recovered by considering a multiplicative LBM characterized by a roughness parameter $ξ+ 4γ^2$ and an intermittency exponent $γ$.

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