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Cédric Bernardin

Publications and source records attributed to Cédric Bernardin.

At least 19 recordsLinked to original sources
arXiv2025

Spiking and Resetting

We consider a one-dimensional piecewise deterministic Markov process (PDMP) on $[0,1]$ with resetting at $0$ and depending on a small parameter $\varepsilon>0$. In the singular vanishing limit $\varepsilon \to 0$ we prove that the `` resetting '' simple point process associated to the PDMP converges to a point process described by a jump Markov process decorated by ``spikes'' distributed as a time-space Poisson point process with intensity proportional to $dt \otimes x^{-2} dx$. This proves rigorously results appeared previously in \cite{SBDKC25} and also justifies partially a conjecture formulated there.

math.PR↗
arXiv2025

Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin

We analyze the \emph{equilibrium fluctuations} of a Hamiltonian chain of oscillators on \(\mathbb{Z}\) with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical \emph{diffusive scaling} \(t \mapsto t n^2\) and an interaction strength tuned by \(n^{-1/2}\), the fluctuation field is known to converge to the \emph{energy solution} of the stochastic Burgers equation (SBE) on the torus~\cite{ABGS22}. We introduce a \emph{coupled moving heat bath} of strength \(n^{-δ}\) acting on the particle system. We prove that for \(δ\leq 1\) (the \emph{strong-coupling regime}), the equilibrium fluctuation field converges to the \emph{energy solution of the SBE with a Dirichlet boundary condition at zero}. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for \(δ> 1\) (the \emph{weak-coupling regime}), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of~\cite{GJ14}. Our analysis thus reveals a sharp \emph{critical scaling} in the coupling strength \(δ\), which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation.

math.PR↗
arXiv2024

To spike or not to spike: the whims of the Wonham filter in the strong noise regime

We study the celebrated Shiryaev-Wonham filter (1964) in its historical setup where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of the authors show the appearance of spikes in the filtered process, akin to a metastability phenomenon. This paper is aimed at understanding the smoothed optimal filter, which is relevant for any system with feedback. In particular, we exhibit a sharp phase transition between a spiking regime and a regime with perfect smoothing.

math.PR↗
arXiv2022

Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion

We study the hydrodynamic limit for a model of symmetric exclusion processes with heavy-tailed long jumps and in contact with infinitely extended reservoirs. We show how the corresponding hydrodynamic equations are affected by the parameters defining the model. The hydrodynamic equations are characterized by a class of super-diffusive operators that are given by the regional fractional Laplacian with some additional reaction terms and various boundary conditions. This work solves some questions left open in \cite{BJGO2} about the same model.

math.PR↗
arXiv2022

Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps

We consider the zero-range process with long jumps and in contact with infinitely many reservoirs in its non-equilibrium stationary state. We derive the hydrostatic limit and the Fick's law, which are a consequence of a relationship between the exclusion process and the zero-range process. We also obtain the large deviation principle for the empirical density, i.e. we compute the non-equilibrium free energy.

math.PR↗
arXiv2021

Emergence of jumps in quantum trajectories via homogeneization

In the strong noise regime, we study the homogeneization of quantum trajectories i.e. stochastic processes appearing in the context of quantum measurement. When the generator of the average semi-group can be separated into three distinct time scales, we start by describing a homogenized limiting semi-group. This result is of independent interest and is formulated outside of the scope of quantum trajectories. Going back to the quantum context, we show that, in the Meyer-Zheng topology, the time-continuous quantum trajectories converge weakly to the discontinuous trajectories of a pure jump Markov process. Notably, this convergence cannot hold in the usual Skorokhod topology.

math.PR↗
arXiv2021

Heat transport in an ordered harmonic chain in presence of a uniform magnetic field

We consider heat transport across a harmonic chain of charged particles, with transverse degrees of freedom, in the presence of a uniform magnetic field. For an open chain connected to heat baths at the two ends we obtain the nonequilibrium Green's function expression for the heat current. This expression involves two different Green's functions which can be identified as corresponding respectively to scattering processes within or between the two transverse waves. The presence of the magnetic field leads to two phonon bands of the isolated system and we show that the net transmission can be written as a sum of two distinct terms attributable to the two bands. Exact expressions are obtained for the current in the thermodynamic limit, for the the cases of free and fixed boundary conditions. In this limit, we find that at small frequency $ω$, the effective transmission has the frequency-dependence $ω^{3/2}$ and $ω^{1/2}$ for fixed and free boundary conditions respectively. This is in contrast to the zero magnetic field case where the transmission has the dependence $ω^2$ and $ω^0$ for the two boundary conditions respectively, and can be understood as arising from the quadratic low frequency phonon dispersion.

cond-mat.stat-mech↗
arXiv2021

Localization effects due to a random magnetic field on heat transport in a harmonic chain

We consider a harmonic chain of $N$ oscillators in presence of a disordered magnetic field. The ends of the chain are connected to heat baths and we study the effects of the magnetic field randomness on the heat transport. The disorder, in general, causes localization of the normal modes due to which a system becomes insulating. However, for this system, the localization length diverges as the normal mode frequency approaches zero. Therefore, the low frequency modes contribute to the transmission, $\mathcal{T}_N(ω)$, and the heat current goes down as a power law with the system size, $N$. This power law is determined by the small frequency behaviour of some Lyapunov exponent, $λ(ω)$, and the transmission in the thermodynamic limit, $\mathcal{T}_\infty(ω)$. While it is known that in the presence of a constant magnetic field $\mathcal{T}_\infty(ω)\sim ω^{3/2},~ ω^{1/2}$ depending on the boundary conditions, we find that the Lyapunov exponent for the system behaves as $λ(ω)\simω$ for $\expval{B}\neq 0$ and $λ(ω)\simω^{2/3}$ for $\expval{B}=0$. Therefore, we obtain different power laws for current vs $N$ depending on $\expval{B}$ and the boundary conditions.

cond-mat.stat-mech↗
arXiv2020

Spiking and collapsing in large noise limits of SDEs

We analyze the strong noise limit of one-dimensional stochastic differential equations (SDEs). Our initial motivation comes from continuous measurements of open quantum systems. In this context, Bauer, Bernard and Tilloy pointed out an intriguing behavior. As the noise grows larger, the solutions exhibit locally a collapsing, that is to say, converge to pure jump processes very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process. We give a precise meaning to the convergence and completely prove these statements for a large class of one-dimensional diffusions, thanks to a robust strategy of proof.

math.PR↗
arXiv2019

Hydrodynamic limit for a disordered harmonic chain

We consider a one-dimensional unpinned chain of harmonic oscillators with random masses. We prove that after hyperbolic scaling of space and time the distributions of the elongation, momentum and energy converge to the solution of the Euler equations. Anderson localization decouples the mechanical modes from the thermal modes, allowing the closure of the energy conservation equation even out of thermal equilibrium. This example shows that the derivation of Euler equations rests primarily on scales separation and not on ergodicity. Furthermore, it follows from our proof that the temperature profile does not evolve in any space-time scale.

math.PR↗
arXiv2018

Fractional diffusion equation description of an open anomalous heat conduction set-up

We provide a stochastic fractional diffusion equation description of energy transport through a finite one-dimensional chain of harmonic oscillators with stochastic momentum exchange and connected to Langevian type heat baths at the boundaries. By establishing an unambiguous finite domain representation of the associated fractional operator, we show that this equation can correctly reproduce equilibrium properties like Green-Kubo formula as well as non-equilibrium properties like the steady state temperature and current. In addition, this equation provides the exact time evolution of the temperature profile. Taking insights from the diffusive system and from numerical simulations, we pose a conjecture that these long-range correlations in the steady state are given by the inverse of the fractional operator. We also point out some interesting properties of the spectrum of the fractional operator. All our analytical results are supplemented with extensive numerical simulations of the microscopic system.

cond-mat.stat-mech↗
arXiv2017

Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence

In [2] it has been proved that a linear Hamiltonian lattice field perturbed by a conservative stochastic noise belongs to the 3/2-Lévy/Diffusive universality class in the nonlinear fluctuating theory terminology [15], i.e. energy superdiffuses like an asymmetric stable 3/2-Lévy process and volume like a Brownian motion. According to this theory this should remain valid at zero tension if the harmonic potential is replaced by an even potential. In this work we consider a quartic anharmonicity and show that the result obtained in the harmonic case persists up to some small critical value of the anharmonicity.

math-ph↗
arXiv2017

Interpolation process between standard diffusion and fractional diffusion

We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume [5, 3]. We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein-Uhlenbeck process driven by a Lévy process which interpolates between Brownian motion and the maximally asymmetric 3/2-stable Lévy process. This result extends and solves a problem left open in [4].

math.PR↗
arXiv2015

Thermal conductivity in harmonic lattices with random collisions

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.

cond-mat.stat-mech↗
arXiv2014

From normal diffusion to superdiffusion of energy in the evanescent flip noise limit

We consider a harmonic chain perturbed by an energy conserving noise depending on a parameter $γ$. When $γ$ is of order one, the energy diffuses according to the standard heat equation after a space-time diffusive scaling. On the other hand, when $γ=0$, the energy superdiffuses according to a 3/4 fractional heat equation after a subdiffusive space-time scaling. In this paper, we study the existence of a crossover between these two regimes as a function of $γ$.

math.PR↗
arXiv2014

Occupation times of long-range exclusion and connections to KPZ class exponents

With respect to a class of long-range exclusion processes on $\mathbb{Z}^d$, with single particle transition rates of order $|\cdot|^{-(d+α)}$, starting under Bernoulli invariant measure $ν_ρ$ with density $ρ$, we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on $α$, $d$ and $ρ$ with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter $H\in [1/2,3/4]$. However, in the asymmetric case, we study the asymptotics of the variances, which when $d=1$ and $ρ=1/2$ points to a curious dichotomy between long-range strength parameters $0<α\leq 3/2$ and $α>3/2$. In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short-range model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.

math.PR↗