This paper studies a class of mixed mean-field jump processes on an abstract state space Π, together with their associated N-particle systems. The dynamics consist of the superposition of an independent Markovian component and a bounded mean-field jump interaction; in particular, piecewise deterministic Markov processes (PDMPs) with mean-field interactions are covered by this framework. Under a second-order bounded difference condition on the mean-field jump kernel, we establish entropic propagation of chaos as N →∞. In particular, we obtain an explicit qualitative bound on the relative entropy between the law of the N-particle system and the product measure induced by the mean-field limit. The proof relies on the second-order concentration inequality introduced in Götze and Sambale, 2020.
The Discrete Ordinates Method (DOM) is widely used for velocity discretization in radiative transport simulations. However, DOM tends to exhibit the ray effect when the velocity discretization is not sufficiently refined, a limitation that is well documented. To counter this, we have developed the Random Ordinates Method (ROM) by integrating randomness into the velocity discretization, which mitigates the ray effect without incurring additional computational costs. ROM partitions the velocity space into n cells, selects a random ordinate from each cell, and solves a DOM system with these ordinates. It leverages the average of multiple samples to achieve a higher convergence order, especially for solutions with low regularity in the velocity variable. In this work, we provide a detailed convergence analysis for ROM, focusing on bias and single-run errors. This analysis is crucial for determining the necessary mesh size and the optimal number of samples required to attain a specified level of accuracy.
We prove that under a generic asymptotic condition on the charge, the small data solutions to the Vlasov-Maxwell system do not verify linear scattering. In other words, we show the non-L^1 asymptotic completeness of the system. The proof makes use of the Lorentz invariance of the equations.
In this paper, we develop an analytical framework for the partial differential equation underlying the consensus-based optimization model. The main challenge arises from the nonlinear, nonlocal nature of the consensus point, coupled with a diffusion term that is both singular and degenerate. By employing a regularization procedure in combination with a compactness argument, we establish the global existence and uniqueness of weak solutions in L^∞(0,T;L^1∩ L^∞(ℝ^d)). Furthermore, we show that the weak solutions exhibit improved H^2-regularity when the initial data is regular.
