arXiv subjects

Manh Hong Duong

Publications and source records attributed to Manh Hong Duong.

At least 19 recordsLinked to original sources

arXiv2026

Gradient flow approach to Landau equation: A cross-product structure

We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form $A(|v-v_*|)\sim \langle v-v_*\rangle^γ|v-v_*|^2$ with $γ\in(-\infty,1]$, this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted $L^1$-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.

math.AP↗

arXiv2026

GENERIC formulation and small-angle limit for Kinetic wave equations

In this paper, we formulate the three-wave and four-wave kinetic equations into the GENERIC framework and formally derive a small-angle limit for the four-wave equation. This limit is akin to the well-known grazing limit from the kinetic Boltzmann equation to the kinetic Landau equation. We also show the GENERIC structure of the limiting system.

math.AP↗

arXiv2026

Trust or Check? Understanding the (Evolutionary) Dynamics of User Trust in AI Systems

As the capabilities and adoption of Artificial Intelligence (AI) systems grow, trust in these AI systems is an increasingly urgent concern. Much research has focused on models of AI governance and has primarily examined incentives for safe development and effective regulation. Hence they typically represented users trust as a one-shot adoption choice rather than as a dynamic, evolving process shaped by repeated interactions. We instead model trust as the dynamic choice of reduced monitoring in a repeated, asymmetric interaction between users and AI developers, where checking developers' behaviour is costly. Using evolutionary game theory, we study how users' strategies of trust and developers' strategies of providing safe (compliant) or unsafe (non-compliant) AI co-evolve under different levels of monitoring cost and institutional regimes. We conduct the analysis on both imitation-based and learning-based perspectives, with the stochastic finite-population dynamics, the infinite-population replicator analysis and the reinforcement learning analysis. We find three robust long-run regimes: no adoption by users while developers provide unsafe AI, unsafe but widely adopted systems, and safe systems that are widely adopted. Only the last is desirable, and it arises when penalties for unsafe behaviour exceed the extra cost of safety and users can still afford to monitor at least occasionally. Our results formally support governance proposals that emphasise transparency, low-cost monitoring, and meaningful sanctions, and they show that neither regulation alone nor blind user trust is sufficient to prevent the drift towards unsafe or low-adoption outcomes.

cs.AI↗

arXiv2026

Co-evolution of social reward and punishment under institutional interventions

We investigate how peer and institutional incentives jointly shape the evolution of cooperation, social welfare, and enforcement efficiency in social dilemmas. In a Prisoners Dilemma with four strategies, unconditional cooperators (C), defectors (D), social punishers (SP), and social rewarders (SR), we allow decentralised peer incentives and centralised institutional incentives to act simultaneously, with the institution able to reward or punish any subset of strategies. In infinite well-mixed populations, we analyse the resulting four-strategy replicator dynamics, and in structured populations we use agent-based simulations on square lattices to study spatial effects and network reciprocity. Intervention schemes are evaluated by equilibrium states and evolutionary flow for infinite well-mixed populations, by cooperation levels and social welfare for structured populations, defined as aggregate population payoff net of institutional cost. We find that peer punishment most strongly promotes cooperation, whereas peer reward is more beneficial for social welfare. Institutionally rewarding peer incentive strategies substantially improves both cooperation and welfare, while subsidising unconditional cooperators has little impact. Under institutional punishment, directly penalising defectors is the only consistently effective policy; punishing peer incentive strategies dismantles decentralised incentives, reduces cooperation, and harms social welfare, showing that maximising cooperation does not necessarily optimise overall societal benefit. Our findings provide design principles for institutions seeking to balance cooperation promotion with welfare maximisation.

cs.AI↗

arXiv2026

Reputation-driven Cooperation in Lattice-based Decentralized Federated Learning through Evolutionary Game Theory

Decentralized Federated Learning (DFL) has emerged as an optimal privacy-preserving solution; however, it remains vulnerable to opportunistic behaviors due to the absence of a central coordinator. While Evolutionary Game Theory (EGT) serves as a powerful framework for analyzing such behaviors, existing studies often assume that agents possess perfect rationality and maintain static strategies. To address these limitations, this paper proposes a novel EGT framework designed to analyze strategic evolution and enhance overall system performance. The primary contributions of this work are threefold: First, we model peer-to-peer (P2P) interactions on a lattice network structure under the assumption of bounded rationality. Second, we formulate a comprehensive payoff matrix incorporating training costs, communication overhead, and cooperative rewards, while tailoring a strategy update rule that captures spatial propagation dynamics. Third, we integrate a reputation-based reward-and-punishment mechanism to effectively deter free-riding behaviors. Simulation results demonstrate that the framework significantly outperforms the baseline. Specifically, it increases average accuracy from approximately 70% to 82%, elevates cooperation frequency to approach 100% (compared to below 5% in the baseline), and drops accuracy variance from around 0.40 to 0.002, thereby accelerating uniform convergence and ensuring system stability.

cs.AI↗

arXiv2026

Global Dynamics of Trait-Structured Generalised Lotka-Volterra Systems with Trait-Independent Interactions

We study the long time dynamics of a selection-mutation integro-differential Lotka-Volterra system of $N$ populations. In our model, fitness depends on a continuous phenotypic trait, but the effect of one population on another is independent of this trait. We establish that, under some usual assumptions on the interactions between populations, the long time behaviour of solutions is exactly determined by the well studied Generalised Lotka-Volterra (ordinary differential) Equation. We also show that, all else being equal, the minimal mutation rate is selected for in a static environment, whereas in an a changing environment, an intermediate mutation rate could be selected instead. The key step in our proofs is establishing that the total population sizes are asymptotically governed by a Generalised Lotka-Volterra Equation, which then allows the use of a general result on asymptotically autonomous dynamical systems.

math.AP↗

arXiv2026

The Homogeneous Landau Equation with Regularised Thermal Noise

We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation--dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-Itô correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.

math.AP↗

arXiv2026

Multi-species McKean-Vlasov dynamics in non-convex landscapes

In this paper, we study multi-species stochastic interacting particle systems and their mean-field McKean-Vlasov partial differential equations (PDEs) in non-convex landscapes. Under general assumptions on non-convex confining and interaction potentials with polynomial growth, we establish the well-posedness of the multi-species SDE system, prove propagation of chaos, deriving the corresponding coupled McKean-Vlasov PDE system in the mean-field limit. Our focus is on the long-time and asymptotic behaviour of the mean-field PDEs. For quadratic interaction potentials and under an appropriate structural assumption, which implies that the generator of each species is multiple of a common generator, we show the existence and (non-) uniqueness of stationary solutions, study their linear stability and prove the existence of a phase transition at low noise strengths. For quadratic and symmetric interaction potentials (but no structural assumption), we construct a free-energy functional that plays the role of a Lyapunov function for the mean-field PDE system. Furthermore, we establish the convergence of solutions to the mean-field PDEs (and of their free energy) to a stationary state (and the corresponding free energy).

math.PR↗

arXiv2026

Social welfare optimisation under institutional reward and punishment

Institutional incentives are widely used to promote cooperation among autonomous, self-regarding agents, from human societies to multi-agent and AI systems. Existing work typically treats incentive design as a bi-objective problem: minimise institutional cost while achieving a high long-run frequency of cooperation. Whether such schemes also maximise social welfare - total population payoff net of institutional expenditure - has remained largely unexplored. We develop a welfare-centric framework for institutional incentives in finite, well-mixed populations playing a social dilemma (Donation Game and Public Goods Game), considering both rewards for cooperators and punishments for defectors. For each mechanism, we derive explicit expressions for expected social welfare and characterise how it depends on incentive efficiency and selection intensity. Analytically, we identify parameter regimes where social welfare has a single optimal incentive level and regimes with qualitative phase transitions, in which welfare becomes non-monotonic with multiple local optima. We prove that any welfare-maximising incentive is either zero or concentrated around a simple closed-form target, and we provide an efficient algorithm to compute these optima. Comparing reward and punishment, we further derive close-formed conditions under which reward outperform punishment in terms of social welfare for any given budget. Overall, our results reveal a systematic gap between incentives optimised for cost or cooperation frequency and those that maximise welfare.

cs.GT↗

arXiv2026

Trend to equilibrium and Newtonian limit for the relativistic Langevin equation with singular potentials

We study a system of interacting particles in the presence of the relativistic kinetic energy, external confining potentials, singular repulsive forces as well as a random perturbation through an additive white noise. In comparison with the classical Langevin equations that are known to be exponentially attractive toward the unique statistically steady states, we find that the relativistic systems satisfy algebraic mixing rates of any order. This relies on the construction of Lyapunov functions adapting to previous literature developed for irregular potentials. We then explore the Newtonian limit as the speed of light tends to infinity and establish the validity of the approximation of the solutions by the Langevin equations on any finite time window.

math.PR↗

arXiv2026

Unified Formulation and Asymptotic Limits of Inhomogeneous Kinetic Models within GENERIC

In this paper, we study a general class of inhomogeneous kinetic models that unifies fundamental models in both the statistical physics of particles and of waves, namely the kinetic Boltzmann equations and the kinetic wave equations, in both classical (non-relativistic), relativistic and quantum settings. We formulate this unified equation into the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework. We then derive the grazing (small-angle) limit in two-body interaction systems, which leads to Landau-type equations. Finally, we show that these limiting systems can also be formulated as GENERIC systems.

math.AP↗

arXiv2026

Asymptotic analysis for the Generalized Relativistic Langevin Equation

In this paper, we study a non-Markovian generalized relativistic Langevin equation (GRLE). We show that when the memory kernel is a sum of exponentials, the GRLE is equivalent to a Markovian system with added variables. We establish the well-posedness and polynomial ergodicity, obtaining an algebraic rate of convergence to the unique Gibbs distribution. From the Markovian GRLE, we recover the relativistic underdamped Langevin dynamics in a small-noise limit, as well as the classical (non-relativistic) generalized Langevin dynamics in the Newtonian limit.

math.PR↗

arXiv2026

Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noises

In this paper, we study systems of $N$ interacting particles described by the classical and relativistic Langevin dynamics with singular forces and multiplicative noises. For the classical model, we prove the ergodicity, obtaining an exponential rate of convergence to the invariant Boltzmann-Gibbs distribution, and the small-mass limit, recovering the $N$-particle interacting overdamped Langevin dynamics. For the relativistic model, we establish the ergodicity, obtaining an algebraic mixing rate of any order to the Maxwell-Jüttner distribution, and the Newtonian limit (that is when the speed of light tends to infinity), approximating a system of underdamped Langevin dynamics. The proofs rely on the construction of Lyapunov functions that account for irregular potentials and multiplicative noises.

math.PR↗

arXiv2026

Multi-species kinetic models: GENERIC formulation and Fisher information

In this paper, we study the GENERIC structures of multi-species spatially inhomogeneous Boltzmann and Landau equations with Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac statistics. In addition, under suitable assumptions on the collision kernels, we show that the Fisher information for the multi-species spatially homogeneous Boltzmann equation is non-increasing in time.

math.AP↗

arXiv2026

On a fuzzy Landau Equation: Part I. A variational approach

This article is the first in a series of works on the fuzzy Landau equation, where particles interact through delocalised Coulomb collisions. Here, we establish a variational characterisation that recasts the fuzzy Landau equation within the framework of GENERIC systems (General Equations for Non-Equilibrium Reversible-Irreversible Coupling).

math.AP↗

arXiv2026

Eco-evolutionary dynamics of a trait-structured predator-prey model

The coupling between evolutionary and ecological changes (eco-evolutionary dynamics) has been shown to be relevant among diverse species, and is also of interest outside of ecology, i.e. in cancer evolution. These dynamics play an important role in determining survival in response to climate change, motivating the need for mathematical models to capture this often complex interplay. Models incorporating eco-evolutionary dynamics often sacrifice analytical tractability to capture the complexity of real systems, do not explicitly consider the effect of population heterogeneity, or focus on long-term behaviour. In order to capture population heterogeneity, both transient, and long-term dynamics, while retaining tractability, we generalise a moment-based method applicable in the regime of small segregational variance to the case of time-dependent mortality and birth. These results are applied to a predator-prey model, where ecological parameters such as the contact rate between species are trait-structured. The trait-distribution of the prey species is shown to be approximately Gaussian with constant variance centered on the mean trait, which is asymptotically governed by an autonomous ODE. In this way, we make explicit the impact of eco-evolutionary dynamics on the transient behaviour and long-term fate of the prey species.

math.AP↗

arXiv2025

On a fuzzy Landau Equation: Part III. The grazing collision limit

In this paper, we study the grazing limit from the non-cutoff fuzzy Boltzmann equations to the fuzzy Landau equation, where particles interact through delocalised collisions. We show the grazing limit through variational formulations that correspond to the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) structure of the respective equations. We show that the variational formulation associated with a non-quadratic dual dissipation pair for the fuzzy Boltzmann equations converges to a variational formulation of the fuzzy Landau equation corresponding to a quadratic dissipation pair.

math.AP↗

arXiv2025

Can Media Act as a Soft Regulator of Safe AI Development? A Game Theoretical Analysis

When developers of artificial intelligence (AI) products need to decide between profit and safety for the users, they likely choose profit. Untrustworthy AI technology must come packaged with tangible negative consequences. Here, we envisage those consequences as the loss of reputation caused by media coverage of their misdeeds, disseminated to the public. We explore whether media coverage has the potential to push AI creators into the production of safe products, enabling widespread adoption of AI technology. We created artificial populations of self-interested creators and users and studied them through the lens of evolutionary game theory. Our results reveal that media is indeed able to foster cooperation between creators and users, but not always. Cooperation does not evolve if the quality of the information provided by the media is not reliable enough, or if the costs of either accessing media or ensuring safety are too high. By shaping public perception and holding developers accountable, media emerges as a powerful soft regulator -- guiding AI safety even in the absence of formal government oversight.

cs.AI↗