Well-Posedness of the Second-Order SDEs Describing an N-Particle System Interacting via Coulomb Interaction

This paper concerns a second-order N interacting stochastic particle system with singular potential for any dimension n ≥ 2. By some estimates of total energy of the system, we prove that there is no collision among particles almost surely in any finite time interval, then the well-posedness of this interacting particle system can be established.

Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation

In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a stochastic particle system while introducing an intermediate particle system with smooth interaction potential. The theoretical results are compared to numerical simulations relying on suitable discretization schemes for the microscopic and macroscopic level. In particular, the regime switch where the analytic theory fails is numerically analyzed very carefully and allows for a better understanding of the equation.

Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

We consider an interacting [Formula: see text]-particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuity-type of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles [Formula: see text] goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak solution to the continuity equation. This extends previous results on an interacting particle system with bounded and Lipschitz continuous drift terms and normal reflecting boundary conditions by Sznitman [J. Funct. Anal. 56 (1984) 311–336] to that one with discontinuous kernels.

Chaos Propagation in Games of Control for Energy-Efficient Wireless Communications

The question of energy management is a major engineering challenge in any wireless communication system that is constrained by finite battery resources. In this paper, we address the problem of transmit power control in a multi-user wireless network with a single base station and random time-varying channels transmitting in parallel and interfering between each other. We suggest a stochastic particle system for studying the game of control; specifically, the channels are modeled as purely discontinuous Markov processes with known characteristics. The signal to interference plus noise ratio (SINR) determines the interactions among particles. Dimensionality reduction of the multi-objective optimization problem relies on chaos propagation, i.e., on the fact that the independence of particles persists in time as and when their number becomes large. And then we solve the optimal control problem for one representative of the particle system. Furthermore, we develop a martingale approach for an alternative open-loop optimal control instead of the feedback one since the latter is not computationally feasible.

Spectral theory for the -Boson particle system

We develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the $q$-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with $q$-TASEP ($q$-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of $q$-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our $q$-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.

VARIATIONAL FORMULATION OF THE FOKKER–PLANCK EQUATION WITH DECAY: A PARTICLE APPROACH

We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the many-particle limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Mosco-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker–Planck equation with decay.