arXiv subjects

Léonard Ferdinand

Publications and source records attributed to Léonard Ferdinand.

7 recordsLinked to original sources

arXiv2026

Stochastic PDE construction of some non-local or non-translation-invariant Anderson operators in the full subcritical regime

We construct Anderson-type operators in the full subcritical regime via a SPDE argument. Our work encompasses perturbations by a Gaussian noise of the fractional Laplacian on the torus or on the full space, and of the harmonic oscillator on the full space. In every case, we are able to derive an optimal estimate on the difference between the solution and the deterministic heat flow, which allows us to get a Weyl law for the fractional case on the torus and for the harmonic oscillator case, and so in the full subcritical regime. For the non-translation-invariant operator, the main novelty is that we develop a general argument in order to prove how to renormalise the operator with some constants. The SPDE argument uses the flow approach introduced by Duch. To construct the operators, we use the Hille-Yosida theorem for the fractional Laplacian on the torus and the harmonic oscillator, and the Klein-Landau theorem for the fractional Laplacian in the full space.

math.PR↗

arXiv2026

Rough differential equations in the flow approach

We show how the flow approach of Duch, with elementary differentials as coordinates, can be used to prove well-posedness for rough stochastic differential equations driven by fractional Brownian motion with Hurst index $H > \frac{1}{4}$. A novelty appearing here is that we use coordinates for the flow that are indexed by trees rather than multi-indices.

math.PR↗

arXiv2025

A flow approach to the generalized KPZ equation

We show that the flow approach of Duch [Duc21] can be adapted to prove local well-posedness for the generalized Kardar-Parisi-Zhang equation. The key step is to extend the flow approach so that it can accommodate semi-linear equations involving smooth, non-polynomial, functions of the solution - this is accomplished by introducing coordinates for the flow built out of elementary differentials.

math.PR↗

arXiv2024

A Stochastic Analysis Approach to Tensor Field Theories

We present two different arguments using stochastic analysis to construct super-renormalizable tensor field theories, namely the $\mathrm{T}^4_3$ and $\mathrm{T}^4_4$ models. The first approach is the construction of a Langevin dynamic combined with a PDE energy estimate while the second is an application of the variational approach of Barashkov and Gubinelli. By leveraging the melonic structure of divergences, regularising properties of non-local products, and controlling certain random operators, we demonstrate that for tensor field theories these arguments can be significantly simplified in comparison to what is required for $Φ^4_d$ models.

math.PR↗

arXiv2023

Spectrally cut-off GFF, regularized $Φ^4$ measure, and reflection positivity

We argue that the spectrally cut-off Gaussian free field $Φ_Λ$ on a compact Riemannian manifold or on $\mathbb{R}^n$ cannot satisfy the spatial Markov property. Moreover, when the manifold is reflection positive, we show that $Φ_Λ$ fails to be reflection positive. We explain the difficulties one encounters when trying to deduce the reflection positivity property of the measure exp$(-\|ρΦ_Λ\|_{L^4}^4) μ_{\text{GFF}}(dΦ)$ from the reflection positivity property of the Gaussian free field measure $μ_{\text{GFF}}$ in a naive way. These issues are probably well-known to experts of constructive quantum field theory but to our knowledge, no detailed account can be found in the litterature. Our pedagogical note aims to fill this small gap.

math.PR↗

arXiv2022

Borel summability of the 1/N expansion in quartic O(N)-vector models

We consider a quartic O(N)-vector model. Using the Loop Vertex Expansion, we prove the Borel summability in 1/N along the real axis of the partition function and of the connected correlations of the model. The Borel summability holds uniformly in the coupling constant, as long as the latter belongs to a cardioid like domain of the complex plane, avoiding the negative real axis.

math-ph↗

arXiv2021

Closer look at white hole remnants

The idea that, after their evaporation, Planck-mass black holes might tunnel into metastable white holes has recently been intensively studied. Those relics have been considered as a dark matter candidate. We show that the model is severely constrained and underline some possible detection paths. We also investigate, in a more general setting, the way the initial black hole mass spectrum would be distorted by both the bouncing effect and the Hawking evaporation.

gr-qc↗