arXiv subjects

Alexandra Holzinger

Publications and source records attributed to Alexandra Holzinger.

6 recordsLinked to original sources

arXiv2026

Well-posedness and mean-field limit of discontinuous weighted dynamics via the relative entropy method

We consider deterministic particle dynamics with time evolving weights and their associated Kolmogorov equation and mean-field equation. We prove existence and unique- ness for the limit PDE alongside estimates on the growth of the logarithmic gradient as well as existence of weak solutions for the Kolmogorov equation satisfying an appropriate entropy inequality. We then apply these estimates and the relative entropy method as developed in [17], in order to derive the associated equation as a mean field limit. Our results cover both interactions and influence kernels with mild regularity assumptions.

math.AP↗

arXiv2025

Propagation of chaos for multi-species moderately interacting particle systems up to Newtonian singularity

We derive a class of multi-species aggregation-diffusion systems from stochastic interacting particle systems via relative entropy method with quantitative bounds. We show an algebraic $L^1$-convergence result using moderately interacting particle systems approximating attractive/repulsive singular potentials up to Newtonian/Coulomb singularities without additional cut-off on the particle level. The first step is to make use of the relative entropy between the joint distribution of the particle system and an approximated limiting aggregation-diffusion system. A crucial argument in the proof is to show convergence in probability by a stopping time argument. The second step is to obtain a quantitative convergence rate to the limiting aggregation-diffusion system from the approximated PDE system. This is shown by evaluating a combination of relative entropy and $L^2$-distance.

math.PR↗

arXiv2024

Fluctuations around the mean-field limit for attractive Riesz potentials in the moderate regime

A central limit theorem is shown for moderately interacting particles in the whole space. The interaction potential approximates singular attractive or repulsive potentials of sub-Coulomb type. It is proved that the fluctuations become asymptotically Gaussians in the limit of infinitely many particles. The methodology is inspired by the classical work of Oelschl\"ager on fluctuations for the porous-medium equation. The novelty in this work is that we can allow for attractive potentials in the moderate regime and still obtain asymptotic Gaussian fluctuations. The key element of the proof is the mean-square convergence in expectation for smoothed empirical measures associated to moderately interacting $N$-particle systems with rate $N^{-1/2-\varepsilon}$ for some $\varepsilon>0$. To allow for attractive potentials, the proof uses a quantitative mean-field convergence in probability with any algebraic rate and a law-of-large-numbers estimate as well as a systematic separation of the terms to be estimated in a mean-field part and a law-of-large-numbers part.

math.PR↗

arXiv2023

Quantitative convergence in relative entropy for a moderately interacting particle system on $\mathbb{R}^d$

This article shows how to combine the relative entropy method by D. Bresch, P.-E. Jabin, and Z. Wang in arXiv:1706.09564, arXiv:1906.04093 and the regularized $L^2(\mathbb{R}^d)$-estimate by Oelschl\"ager (Probability theory and related fields, 1987) to prove a strong propagation of chaos result for the viscous porous medium equation from a moderately interacting particle system in $L^\infty(0,T; L^1(\mathbb{R}^d))$-norm. In the moderate interacting setting, the interacting potential is a smoothed Dirac Delta distribution, however, current results regarding the relative entropy methods for singular potentials do not apply. The result holds on $\mathbb{R}^d$ for any dimension $d\geq 1$ and provides a quantitative result where the rate of convergence depends on the moderate scaling parameter and the dimension $d\geq 1$. Additionally, the presented method can be adapted for moderately interacting systems for which a certain convergence probability holds -- thus a propagation of chaos result in relative entropy can be obtained for kernels approximating Coulomb potentials.

math.AP↗

arXiv2021

Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems

Population cross-diffusion systems of Shigesada-Kawasaki-Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

math.AP↗

arXiv2021

Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger's approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo--Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.

math.AP↗