arXiv subjects

Aurélien Deya

Publications and source records attributed to Aurélien Deya.

At least 19 recordsLinked to original sources

arXiv2026

On the parabolic $Φ_3^4$ model for the harmonic oscillator II: global existence and invariant measures

We establish an a priori bound for the dynamical parabolic $Φ_3^4$ model with harmonic potential. This bound yields the global well-posedness of the equation and, via the Krylov-Bogoliubov method, the existence of an invariant measure, shown to be non-Gaussian. The argument builds on the strategy developed by Mourrat and Weber for the periodic $Φ_3^4$ model, with substantial modifications to handle the non-compact geometry of $\mathbb{R}^3$ and the spectral framework imposed by the harmonic oscillator. We further prove that this measure is unique in the small-coupling regime.

math.AP↗

arXiv2025

Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise

The study is devoted to the interpretation and wellposedness of the stochastic NLS model \begin{equation*} (\imath \partial_t-Δ)u=|u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T}, \end{equation*} where $\dot{B}$ stands for a space-time fractional noise with index $H=(H_0,H_1)$ in a subset of $(0,1)^{2}$. We first establish that in the situation where $0<2H_0+H_1\leq 2$, the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac74<2H_0+H_1\leq 2$. In this specific case, we exhibit an \textit{explicit} renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.

math.AP↗

arXiv2025

On the 1d stochastic Schr{ö}dinger product

We exhibit various restrictions about the wellposedness of the Schr{\''o}dinger product $$\cl:z \longmapsto -\imath \int\_0^t e^{\imath s {\cop \partial^2\_x}}\big( z\_s\cdot Ψ\_s\big) ds $$ where $Ψ$ refers to the so-called linear solution of the stochastic Schr{\''o}dinger problem. We focus more specifically on the case where $Ψ$ satisfies \begin{equation}\label{starting-equation-abstract} (\imath \partial\_t-\partial^2\_x)Ψ=\dot{B}, \quad Ψ\_0=0,\quad \quad t\in \R, \ x\in \mathbb{T}, \end{equation} where $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac12$. \smallskip As an consequence of our analysis, we obtain that if $H$ is close to $\frac12$ (that is $\dot{B}$ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem \begin{equation*} (\imath \partial\_t-\partial^2\_x)u= |u|^2+\dot{B}, \quad u\_0=0,\quad \quad t\in \R, \ x\in \mathbb{T}, \end{equation*} using only a first-order expansion of the solution (\enquote{$u=Ψ+z$}).

math.AP↗

arXiv2025

On the parabolic $Φ_3^4$ model for the harmonic oscillator: diagrams and local existence

We prove the local wellposedness of the (renormalized) parabolic $Φ^4_3$ model associated with the harmonic oscillator on $\mathbb{R}^3$, that is, the equation formally written as \begin{equation*} \partial_t X + HX= -X^3+\infty\cdot X + ξ, \quad t>0, \quad x \in \mathbb{R}^3, \end{equation*} where $H:=-Δ_{\mathbb{R}^3} +|x|^2$ and $ξ$ denotes a space-time white noise. This model is closely related to the Gross-Pitaevskii equation which is used in the description of Bose-Einstein condensation. Our overall formulation of the problem, based on the so-called paracontrolled calculus, follows the strategy outlined by Mourrat and Weber for the $Φ^4_3$ model on the three-dimensional torus. Significant effort is then required to adapt, within the framework imposed by the harmonic oscillator, the key tools that contribute to the success of this method-particularly the construction of stochastic diagrams at the core of the dynamics.

math.PR↗

arXiv2022

Hyperbolic Anderson model 2: Strichartz estimates and Stratonovich setting

We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz type estimates for the wave kernel in weighted Besov spaces, by which we can prove the wellposedness of an associated Young-type equation. Those Strichartz bounds are of independent interest.

math.PR↗

arXiv2021

On ill-posedness of nonlinear stochastic wave equations driven by rough noise

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide an interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index $H\leq \frac{1}{4}$. Our results clarify and extend previous similar properties exhibited in [3] or in [14].

math.PR↗

arXiv2021

Solving the hyperbolic Anderson model 1: Skorohod setting

This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.

math.PR↗

arXiv2021

A non-linear wave equation with fractional perturbation

We study a $d$-dimensional wave equation model ($2\leq d\leq 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_0,\ldots,H_d) \in (0,1)^{d+1}$ of the noise: if $\sum_{i=0}^d H_i > d-\frac12$, then the equation can be treated directly, while in the case $d-\frac34<\sum_{i=0}^d H_i\leq d-\frac12$, the model must be interpreted in the Wick sense, through a renormalization procedure. Our arguments essentially rely on a fractional extension of the considerations of \cite{gubinelli-koch-oh} for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.

math.PR↗

arXiv2021

A full discretization of the rough fractional linear heat equation

We study a full discretization scheme for the stochastic linear heat equation \begin{equation*}\begin{cases}\partial_t \langleΨ\rangle = Δ\langleΨ\rangle +\dot{B}\, , \quad t\in [0,1], \ x\in \mathbb{R},\\ \langleΨ\rangle_0=0\, ,\end{cases}\end{equation*} when $\dot{B}$ is a very \emph{rough space-time fractional noise}. The discretization procedure is divised into three steps: $(i)$ regularization of the noise through a mollifying-type approach; $(ii)$ discretization of the (smoothened) noise as a finite sum of Gaussian variables over rectangles in $[0,1]\times \mathbb{R}$; $(iii)$ discretization of the heat operator on the (non-compact) domain $[0,1]\times \mathbb{R}$, along the principles of Galerkin finite elements method. We establish the convergence of the resulting approximation to $\langleΨ\rangle$, which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.

math.PR↗

arXiv2020

On stochastic calculus with respect to q-Brownian motion

Following the approach and the terminology introduced in [A. Deya and R. Schott, On the rough paths approach to non-commutative stochastic calculus, J. Funct. Anal., 2013], we construct a product L{é}vy area above the $q$-Brownian motion (for $q\in [0,1)$) and use this object to study differential equations driven by the process.We also provide a detailled comparison between the resulting "rough" integral and the stochastic "It{ô}" integral exhibited by Donati-Martin in [C. Donati-Martin, Stochastic integration with respect to $q$ Brownian motion, Probab. Theory Related Fields, 2003].

math.PR↗

arXiv2020

Skorohod and rough integration with respect to the non-commutative fractional Brownian motion

We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the presence of the NC-fBm, at least for a Hurst index H > 1 4.This finally puts us in a position to state and prove the desired comparison result, which can be regarded as an It{ô}-Stratonovich correction formula for the NC-fBm.

math.PR↗

arXiv2020

Moment estimates for some renormalized parabolic Anderson models

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) = \frac12 Δu_{t}(x) + u_{t}(x) \, \dot W_{t}(x)$, for $t\in\mathbb{R}_{+}$ and $x\in \mathbb{R}^{d}$, where $\dot W_{t}(x)$ is a Gaussian noise whose space time covariance function is singular. In this rough context, we shall give some information about the moments of $u_{t}(x)$ when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.

math.PR↗

arXiv2020

A $K$-rough path above the space-time fractional Brownian motion

We construct a $K$-rough path above either a space-time or a spatial fractional Brownian motion, in any space dimension $d$. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.

math.PR↗

arXiv2020

A nonlinear Schr{ö}dinger equation with fractional noise

We study a stochastic Schr{ö}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schr{ö}dinger group, which allows us to cope with the strong irregularity of the solution.

math.AP↗

arXiv2018

Integration with respect to the Hermitian fractional Brownian motion

For every $d\geq 1$, we consider the $d$-dimensional Hermitian fractional Brownian motion (HfBm), that is the process with values in the space of $(d\times d)$-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the approach of [A. Deya and R. Schott: On the rough paths approach to non-commutative stochastic calculus, JFA (2013)] to define a natural integral with respect to the HfBm when $H>\frac13$, and identify this interpretation with the rough integral with respect to the $d^2$ entries of the matrix. Using this correspondence, we establish a convenient It{ô}--Stratonovich formula for the Hermitian Brownian motion. Finally, we show that at least when $H\geq \frac12$, and as the size $d$ of the matrix tends to infinity, the integral with respect to the HfBm converges (in the tracial sense) to the integral with respect to the so-called non-commutative fractional Brownian motion.

math.PR↗

arXiv2018

Integration with respect to the non-commutative fractional Brownian motion

We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian in a non-commutative probability setting.When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an It{ô}-type approach is known to be possible.When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined L{é}vy area process. We show that such an object can indeed be \enquote{canonically} constructed for any $H\in (\frac14,\frac12)$. Finally, when $H\leq 1/4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) L{é}vy area above the standard fractional Brownian.

math.PR↗

arXiv2018

A priori estimates for rough PDEs with application to rough conservation laws

We introduce a general weak formulation for PDEs driven by rough paths, as well as a new strategy to prove well-posedness. Our procedure is based on a combination of fundamental a priori estimates with (rough) Gronwall-type arguments. In particular this approach does not rely on any sort of transformation formula (flow transformation, Feynman--Kac representation formula etc.) and is therefore rather flexible. As an application, we study conservation laws driven by rough paths establishing well--posedness for the corresponding kinetic formulation.

math.AP↗

arXiv2017

On a non-linear 2D fractional wave equation

We pursue the investigations initiated in [Aur{é}lien Deya: A non-linear wave equation with fractional perturbation (2017)] about a wave-equation model with quadratic perturbation and stochastic forcing given by a space-time fractional noise. We focus here on the two-dimensional situation and therein extend the results of the previous reference to a rougher noise, through the use of a third-order expansion. We also point out the limits of the Wick-renormalisation procedure in this case.

math.PR↗