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Arnaud Debussche

Publications and source records attributed to Arnaud Debussche.

At least 19 recordsLinked to original sources
arXiv2026

Coupled stochastic variational principles for multiscale surface gravity waves -- Part I: theoretical framework

This paper develops a comprehensive stochastic variational framework for surface gravity waves. Starting from Luke's variational principle for irrotational, incompressible free-surface flow, we introduce a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic noise term representing unresolved scales. A path-wise variational principle yields the stochastic counterparts of the Laplace, Bernoulli, and kinematic boundary conditions. To close the system, a second variational principle in expectation is formulated, providing evolution equations for the noise correlation functions. The resulting coupled system preserves the Hamiltonian structure of the Zakharov--Craig--Sulem formulation. We further analyze explicit solutions via WKB approximation and show that the noise correlation functions satisfy a Hamilton-Jacobi equation with ray-tracing dynamics. This framework provides a rigorous foundation for reduced stochastic models of ocean waves and for the kinetic theory developed in the companion paper.

math-ph↗
arXiv2026

Fluctuation dynamics in randomly advected Navier-Stokes equations below critical scaling

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $\delta$; the critical regime $\varepsilon = \delta$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\varepsilon = o (\delta)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\varepsilon = o (\delta^{1 + \iota})$ for some $\iota > 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

math.AP↗
arXiv2026

Law of large numbers for stochastic multiscale spatial gene networks

We study a stochastic multiscale spatial gene network. These naturally arise in molecular biology. In our model, the reactants are subject to on-site reactions on both scales and diffusion on the continuous scale only, although diffusion on both scales could easily be handled. We obtain, under a light condition on the scales between the total population size and the mesh discretisation, the convergence of the stochastic system to a deterministic system consisting of a PDE coupled to a ODE. This is in contrast with the well-stirred case where jumps remain at the limit. In order to prove this convergence result, we develop some moments control for martingales in discrete Sobolev topologies and use products rule in discrete Sobolev spaces.

math.PR↗
arXiv2026

Interpretation of stochastic primitive equations with relaxed hydrostatic assumption as a higher order approximation of 3D stochastic Navier-Stokes

In this paper, we investigate the convergence of solutions of a stochastic representation of the three-dimensional Navier-Stokes equations to those of their primitive equations counterpart. Our analysis covers both weak and strong convergence regimes, corresponding respectively to rigid-lid and "fully periodic" boundary conditions. Furthermore, we explore the impact of relaxing the hydrostatic assumption in the stochastic primitive equations by retaining martingale terms as deviations from hydrostatic equilibrium. This modified model, obtained through a specific asymptotic scaling accessible only within the stochastic framework, captures non-hydrostatic effects while remaining within the primitive equations formalism. The resulting generalized hydrostatic model has been shown to be well-posed when the additional terms are regularized using a suitable filter for divergence-free noises under suitable assumptions. Within this setting, we demonstrate that the model provides a higher-order approximation of the 3D Navier-Stokes equations for appropriately scaled noises.

math.AP↗
arXiv2025

Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs

For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach.

math.NA↗
arXiv2025

Stochastic interpretations of the oceanic primitive equations with relaxed hydrostatic assumptions

In this paper, we investigate how weakening the classical hydrostatic balance hypothesis impacts the well-posedness of the stochastic LU primitive equations. The models we consider are intermediate between the incompressible 3D LU Navier-Stokes equations and the LU primitive equations with standard hydrostatic balance. As such, they are expected to be numerically tractable, while accounting well for phenomena within the grey zone between hydrostatic balance and non-hydrostatic processes. Our main result is the well-posedness of a low-pass filtering-based stochastic interpretation of the LU primitive equations, with rigid-lid type boundary conditions, in the limit of ``quasi-barotropic'' flow. This assumption is linked to the structure assumption proposed in the work of Agresti et al., which can be related to the dynamical regime where the primitive equations remain valid. Furthermore, we present and study two eddy-(hyper)viscosity-based models.

math.AP↗
arXiv2024

Periodic nonlinear Schrödinger equation with distributional potential and invariant measures

In this paper, we continue some investigations on the periodic NLSE started by Lebowitz, Rose and Speer and by Bourgain with the addition of a distributional multiplicative potential. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measure, after suitable truncation in the focusing case. The set and the measure are invariant under the flow. The main ingredients used are Strichartz estimates on periodic NLS with distributional potential to obtain local well-posedness for low regularity initial data.

math.AP↗
arXiv2024

Derivation of stochastic models for coastal waves

In this paper, we consider a stochastic nonlinear formulation of classical coastal waves models under location uncertainty (LU). In the formal setting investigated here, stochastic versions of the Serre-Green- Nagdi, Boussinesq and classical shallow water wave models are obtained through an asymptotic expansion, which is similar to the one operated in the deterministic setting. However, modified advection terms emerge, together with advection noise terms. These terms are well-known features arising from the LU formalism, based on momentum conservation principle.

math.PR↗
arXiv2024

From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schr{ö}dinger Equation

We study the convergence of a Zakharov system driven by a time white noise, colored in space, to a multiplicative stochastic nonlinear Schr{ö}dinger equation, as the ion-sound speed tends to infinity. In the absence of noise, the conservation of energy gives bounds on the solutions, but this evolution becomes singular in the presence of the noise. To overcome this difficulty, we show that the problem may be recasted in the diffusion-approximation framework, and make use of the perturbed test-function method. We also obtain convergence in probability. The result is limited to dimension one, to avoid too much technicalities. As a prerequisite, we prove the existence and uniqueness of regular solutions of the stochastic Zakharov system.

math.AP↗
arXiv2024

Variational principles for fully coupled stochastic fluid dynamics across scales

This work investigates variational frameworks for modeling stochastic dynamics in incompressible fluids, focusing on large-scale fluid behavior alongside small-scale stochastic processes. The authors aim to develop a coupled system of equations that captures both scales, using a variational principle formulated with Lagrangians defined on the full flow, and incorporating stochastic transport constraints. The approach smooths the noise term along time, leading to stochastic dynamics as a regularization parameter approaches zero. Initially, fixed noise terms are considered, resulting in a generalized stochastic Euler equation, which becomes problematic as the regularization parameter diminishes. The study then examines connections with existing stochastic frameworks and proposes a new variational principle that couples noise dynamics with large-scale fluid motion. This comprehensive framework provides a stochastic representation of large-scale dynamics while accounting for fine-scale components. Our main result is that the evolution of the small-scale velocity component is governed by a linearized Euler equation with random coefficients, influenced by large-scale transport, stretching, and pressure forcing.

physics.flu-dyn↗
arXiv2024

Some properties of a non-hydrostatic stochastic oceanic primitive equations model

In this paper, we study how relaxing the classical hydrostatic balance hypothesis affects theoretical aspects of the LU primitive equations well-posedness. We focus on models that sit between incompressible 3D LU Navier-Stokes equations and standard LU primitive equations, aiming for numerical manageability while capturing non-hydrostatic phenomena. Our main result concerns the well-posedness of a specific stochastic interpretation of the LU primitive equations. This holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of z. In fact these conditions can be related to the dynamical regime in which the primitive equations remain valid. Moreover, under these conditions, we show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise, thus providing a physical coherence to the LU stochastic model.

math.PR↗
arXiv2023

Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction

We consider systems of damped wave equations with a state-dependent damping coefficient and perturbed by a Gaussian multiplicative noise. Initially, we investigate their well-posedness, under quite general conditions on the friction. Subsequently, we study the validity of the so-called Smoluchowski-Kramers diffusion approximation. We show that, under more stringent conditions on the friction, in the small-mass limit the solution of the system of stochastic damped wave equations converges to the solution of a system of stochastic quasi-linear parabolic equations. In this convergence, an additional drift emerges as a result of the interaction between the noise and the state-dependent friction. The identification of this limit is achieved by using a suitable generalization of the classical method of perturbed test functions, tailored to the current infinite dimensional setting.

math.PR↗
arXiv2023

Rough analysis of two scale systems

We address a slow-fast system of coupled three dimensional Navier--Stokes equations where the fast component is perturbed by an additive Brownian noise. By means of the rough path theory, we establish the convergence in law of the slow component towards a Navier--Stokes system with an It{\^o}--Stokes drift and a rough path driven transport noise. This gives an alternative, more general and direct proof to \cite{DP22}. Notably, the limiting rough path is identified as a geometric rough path, which does not necessarily coincide with the Stratonovich lift of the Brownian motion.

math.PR↗
arXiv2023

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (2019) with a sub-quadratic nonlinearity.

math.AP↗
arXiv2022

A consistent stochastic large-scale representation of the Navier-Stokes equations

In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier-Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier-Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension $3$, to a solution of the deterministic Navier-Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier-Stokes equations as a consistent large-scale model of the deterministic Navier-Stokes equation.

math.AP↗
arXiv2022

Second order perturbation theory of two-scale systems in fluid dynamics

In the present paper we study slow-fast systems of coupled equations from fluid dynamics, where the fast component is perturbed by additive noise. We prove that, under a suitable limit of infinite separation of scales, the slow component of the system converges in law to a solution of the initial equation perturbed with transport noise, and subject to the influence of an additional It\=o-Stokes drift. The obtained limit equation is very similar to turbulent models derived heuristically. Our results apply to the Navier-Stokes equations in dimension $d=2,3$; the Surface Quasi-Geostrophic equations in dimension $d=2$; and the Primitive equations in dimension $d=2,3$.

math.PR↗
arXiv2021

Push-forward method for piecewise deterministic biochemical simulations

A biochemical network can be simulated by a set of ordinary differential equations (ODE) under well stirred reactor conditions, for large numbers of molecules, and frequent reactions. This is no longer a robust representation when some molecular species are in small numbers and reactions changing them are infrequent. In this case, discrete stochastic events trigger changes of the smooth deterministic dynamics of the biochemical network. Piecewise-deterministic Markov processes (PDMP) are well adapted for describing such situations. Although PDMP models are now well established in biology, these models remain computationally challenging. Previously we have introduced the push-forward method to compute how the probability measure is spread by the deterministic ODE flow of PDMPs, through the use of analytic expressions of the corresponding semigroup. In this paper we provide a more general simulation algorithm that works also for non-integrable systems. The method can be used for biochemical simulations with applications in fundamental biology, biotechnology and biocomputing.This work is an extended version of the work presented at the conference CMSB2019.

q-bio.QM↗