arXiv subjects

Hung D. Nguyen

Publications and source records attributed to Hung D. Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction

We consider a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. Our main study is the long time statistics of the system in the singular regime as the memory kernel collapses to a Dirac function. Specifically, we show that provided that sufficiently many directions in the phase space are stochastically forced, there is a unique invariant probability measure to which the system converges, with respect to a suitable Wasserstein-type topology, and at an exponential rate which is independent of the decay rate of the memory kernel. We then prove the convergence of this unique statistically steady state to the unique invariant probability measure of the classical stochastic reaction-diffusion equation in the zero-memory limit. Consequently, we establish the global-in-time validity of the short memory approximation.

math.PR↗

arXiv2026

Finite element approximations of the stochastic Benjamin-Bona-Mahony equation with multiplicative noise

This paper is devoted to the numerical analysis of a fully discrete finite element approximation for the stochastic Benjamin-Bona-Mahony equation driven by multiplicative noise. We first establish the existence and uniqueness of solutions to the stochastic BBM equation within an appropriate variational framework and derive several stability estimates for the continuous problem, including an exponential stability result. For the numerical approximation, a conforming finite element method is employed for spatial discretization and is coupled with the implicit Euler-Maruyama scheme for time integration. The convergence of the fully discrete scheme is investigated under two different classes of multiplicative noise. When the noise coefficient is bounded, we obtain optimal strong error estimates in full expectation by combining exponential stability properties of both the stochastic BBM solution and its fully discrete counterpart with a stochastic Gronwall inequality. In the case of general multiplicative noise, where boundedness assumptions are no longer valid, a localization technique based on high-probability events in the sample space is introduced, leading to sub-optimal convergence rates in probability. Finally, numerical experiments are presented to corroborate the theoretical results and to demonstrate the performance of the proposed method.

math.NA↗

arXiv2025

Polynomial mixing for the stochastic Schrödinger equation with large damping in the whole space

We study the long-time mixing behavior of the stochastic nonlinear Schrödinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate of convergence toward equilibrium has remained unknown. In this work, we address the mixing property in the regime of large damping and establish that solutions are attracted toward the unique invariant probability measure at polynomial rates of arbitrary order. Our approach is based on a coupling strategy with pathwise Strichartz estimates.

math.PR↗

arXiv2025

UAV-Assisted Downlink Satellite Covert Communication

This paper investigates the use of an unmanned aerial vehicle (UAV) to assist covert communication between a low-Earth orbit (LEO) satellite and a ground user under the surveillance of a passive warden. The UAV simultaneously serves its own ground network and acts as a friendly jammer to enhance the covertness of satellite transmissions. We derive a closed-form lower bound on the warden's average minimum detection error probability which is then used to define the covert constraint. Building on this, we formulate an optimization problem to jointly design the UAV's 3D placement, its power allocation, and the satellite's transmit power to maximize the system's covert rate. To solve the resulting non-convex problem, we propose an algorithm based on the block coordinate descent (BCD) and successive convex approximation (SCA) techniques, and further develop a Dinkelbach's algorithm for a special case. Numerical results validate the tightness of the derived bound and demonstrate the effectiveness of the proposed algorithms in configuring optimal system parameters.

eess.SP↗

arXiv2025

Fully discrete finite element methods for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise

We investigate a fully discrete finite element approximation for the stochastic Kuramoto-Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler-Maruyama scheme in time. Rigorous error estimates are established for two distinct noise regimes. In the case of bounded multiplicative noise, we prove optimal strong convergence rates in full expectation. The analysis relies crucially on a stochastic Gronwall inequality and an exponential stability estimate for the PDE solution, which together control the interplay between the nonlinear drift and the multiplicative stochastic forcing. For general multiplicative noise, where boundedness no longer holds, we derive sub-optimal convergence rates in probability by introducing a localization technique based on carefully constructed subsets of the sample space. This dual framework demonstrates that the proposed fully discrete scheme achieves strong convergence under bounded noise and probabilistic convergence under general multiplicative noise, thus providing the first comprehensive error analysis for numerical approximations of the stochastic Kuramoto-Sivashinsky equation.

math.NA↗

arXiv2025

Geometric ergodicity of a stochastic Hamiltonian system

We study the long time statistics of a two-dimensional Hamiltonian system in the presence of Gaussian white noise. While the original dynamics is known to exhibit finite time explosion, we demonstrate that under the impact of the stochastic forcing as well as a deterministic perturbation, the solutions are exponentially attractive toward the unique invariant probability measure. This extends previously established results in which the system is shown to be noise-induced stable in the sense that the solutions are bounded in probability.

math.PR↗

arXiv2025

Exponential mixing for the stochastic Kuramoto-Sivashinsky equation on the 1D torus

In this paper, we study the large-time behaviors of the Kuramoto-Sivashinsky equation (KSE) on the 1D torus while being subjected to random perturbation via additive Gaussian noise. It is well-known that under suitable assumptions on the stochastic forcing, the KSE admits a unique invariant probability measure. In this work, we make further progress on the topic of ergodicity by addressing the issue of convergence rate toward equilibrium. In comparison with the previous results, we can prove that the unique invariant probability measure is exponentially attractive and smallness condition of anti-diffusion coefficient is not necessary here. The proof relies on a coupling argument while making use of Lyapunov functions motivated by those of deterministic equations.

math.PR↗

arXiv2025

Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation

We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way.

math.PR↗

arXiv2024

Exponential mixing in a hydrodynamic pilot--wave theory with singular potentials

We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random perturbations through additive Gaussian noise. Under suitable assumptions on the singularities, we show that the walker dynamics is exponentially attracted toward the unique invariant probability measure. The proof relies on a combination of the Lyapunov technique and an asymptotic coupling specifically tailored to our setting. We also present examples of invariant measures, as obtained from numerical simulations of the walker in two-dimensional Coulomb potentials. Our results extend previous work on the ergodicity of stochastic pilot-wave dynamics established for smooth confining potentials.

math.PR↗

arXiv2024

The inviscid limit for long time statistics of the one-dimensional stochastic Ginzburg-Landau equation

We consider the long time statistics of a one-dimensional stochastic Ginzburg-Landau equation with cubic nonlinearity while being subjected to random perturbations via an additive Gaussian noise. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we find that the dynamics is attractive toward the unique invariant probability measure with a polynomial rate that is independent of the vanishing viscosity. This relies on a coupling technique exploiting a Foias-Prodi argument specifically tailored to the system. Then, in the inviscid regime, we show that the sequence of invariant measures converges toward the invariant measure of the stochastic Schrödinger equation in a suitable Wasserstein distance. Together with the uniform polynomial mixing, we obtain the validity of the inviscid limit for the solutions on the infinite time horizon with a log log rate.

math.PR↗

arXiv2024

Asymptotic analysis for the generalized Langevin equation with singular potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori-Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard-Jones and Coulomb functions.

math.PR↗

arXiv2024

The invariant measure of a walking droplet in hydrodynamic pilot-wave theory

We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential.

math.PR↗

arXiv2024

Stability and invariant measure asymptotics in a model for heavy particles in rough turbulent flows

We study a system of Skorokhod stochastic differential equations (SDEs) modeling the pairwise dispersion (in spatial dimension $d=2$) of heavy particles transported by a rough self-similar, turbulent flow with Hölder exponent $h\in (0,1)$. Under the assumption that $h>0$ is sufficiently small, we use Lyapunov methods and control theory to show that the Markovian system is nonexplosive and has a unique, exponentially attractive invariant probability measure. Furthermore, our Lyapunov construction is radially sharp and gives partial confirmation on a predicted asymptotic behavior with respect to the Hölder exponent $h$ of the invariant probability measure. A physical interpretation of the asymptotics is that intermittent clustering is weakened when the carrier flow is sufficiently rough.

math.PR↗

arXiv2023

Polynomial mixing of a stochastic wave equation with dissipative damping

We study the long time statistics of a class of semi--linear wave equations modeling the motions of a particle suspended in continuous media while being subjected to random perturbations via an additive Gaussian noise. By comparison with the nonlinear reaction settings, of which the solutions are known to possess geometric ergodicity, we find that, under the impact of nonlinear dissipative damping, the mixing rate is at least polynomial of any order. This relies on a combination of Lyapunov conditions, the contracting property of the Markov transition semigroup as well as the notion of $d$--small sets.

math.PR↗

arXiv2023

A Convergence Predictor Model for Consensus-based Decentralised Energy Markets

This letter introduces a convergence prediction model (CPM) for decentralized market clearing mechanisms. The CPM serves as a tool to detect potential cyber-attacks that affect the convergence of the consensus mechanism during ongoing market clearing operations. In this study, we propose a successively elongating Bayesian logistic regression approach to model the probability of convergence of real-time market mechanisms. The CPM utilizes net-power balance among all the prosumers/market participants as a feature for convergence prediction, enabling a low-dimensional model to operate efficiently for all the prosumers concurrently. The results highlight that the proposed CPM has achieved a net false rate of less than 0.01% for a stressed dataset.

eess.SY↗

arXiv2023

The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction--diffusion equation. This together with uniform geometric ergodcity implies the validity of the small mass limit for the solutions on the infinite time horizon $[0,\infty)$, thereby extending previously known results established for the damped wave equations under Lipschitz nonlinearities.

math.PR↗

arXiv2023

Gibbsian dynamics and the generalized Langevin equation

We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of the GLE using a Gibbsian framework, which does not rely on existing Markovian approximations. Moreover, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states, generalizing previous known results for the GLE under particular kernels as a sum of exponentials.

math.PR↗

arXiv2022

Ergodicity of a nonlinear stochastic reaction-diffusion equation with memory

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we study statistically steady states of the system and find that they possess regularities compatible with those of the weak solutions. Moreover, if sufficiently many directions in the phase space are stochastically forced, we employ the \emph{generalized coupling} approach to establish the existence and uniqueness of the invariant probability measure to which the system is exponentially attractive. This extends ergodicity results previously established in [Bonaccorsi et al., SIAM J. Math. Anal., 44 (2012)].

math.PR↗