Related papers: Uniform contractivity in Wasserstein metric for th…
We use the Fourier based Gabetta-Toscani-Wennberg (GTW) metric $d_2$ to study the rate of convergence to equilibrium for the Kac model in $1$ dimension. We take the initial velocity distribution of the particles to be a Borel probability…
In this paper we study Kac's 1D particle system, consisting of the velocities of $N$ particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with…
We study a broad class of high-dimensional mean-field exchange models, encompassing both noisy and singular dynamics, along with their dual processes. This includes a generalized version of the averaging process as well as some…
This paper considers the space homogenous Boltzmann equation with Maxwell molecules and arbitrary angular distribution. Following Kac's program, emphasis is laid on the the associated conservative Kac's stochastic $N$-particle system, a…
We analyze the one dimensional Cucker-Smale (in short CS) model with a weak singular communication weight $\psi(x) = |x|^{-\beta}$ with $\beta \in (0,1)$. We first establish a global-in-time existence of measure-valued solutions to the…
We study the long-time behaviour of both the classical second-order Langevin dynamics and the nonlinear second-order Langevin dynamics of McKean-Vlasov type. By a coupling approach, we establish global contraction in an $L^1$ Wasserstein…
By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for L\'evy processes, we obtain explicit exponential contraction rates in terms of the standard $L^1$-Wasserstein distance for…
Fix a strictly positive measure $W$ on the $d$-dimensional torus $\bb T^d$. For an integer $N\ge 1$, denote by $W^N_x$, $x=(x_1, ..., x_d)$, $0\le x_i <N$, the $W$-measure of the cube $[x/N, (x+\mb 1)/N)$, where $\mb 1$ is the vector with…
We analyze a special class of 1-D quantum walks (QWs) realized using optical multi-ports. We assume non-perfect multi-ports showing errors in the connectivity, i.e. with a small probability the multi- ports can connect not to their nearest…
We consider an analogue of the Kac random walk on the special orthogonal group $SO(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\bR^N$. We obtain sharp asymptotics for the rate of convergence in…
We consider Kac's random walk on $n$-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on $\mathit{SO}(n)$…
We consider Kac's 1D N-particle system coupled to an ideal thermostat at temperature T, introduced by Bonetto, Loss, and Vaidyanathan in 2014. We obtain a propagation of chaos result for this system, with explicit and uniform-in-time rates…
This work deals with a class of one-dimensional measure-valued kinetic equations, which constitute extensions of the Kac caricature. It is known that if the initial datum belongs to the domain of normal attraction of an \alpha-stable law,…
In this paper, a risk-aware motion control scheme is considered for mobile robots to avoid randomly moving obstacles when the true probability distribution of uncertainty is unknown. We propose a novel model predictive control (MPC) method…
An explicit estimate is derived for Kac's mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model based on different numbers of particles. For…
We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool…
We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka…
We study fluctuations of mean-field interacting particle systems around their McKean--Vlasov limit. Our main result provides a uniform-in-time quantitative central limit theorem for the fluctuation process, with convergence rate of order…
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry.…
This paper focuses on the performance and the robustness analysis of stochastic jump linear systems. The state trajectory under stochastic jump process becomes random variables, which brings forth the probability distributions in the system…