arXiv subjects

Robert Alexander Crowell

Publications and source records attributed to Robert Alexander Crowell.

5 recordsLinked to original sources

arXiv2025

Emergence of regularity for limit points of McKean-Vlasov particle systems

The empirical measure of an interacting particle system is a purely atomic random probability measure. In the limit as the number of particles grows to infinity, we show for McKean-Vlasov systems with common noise that this measure becomes absolutely continuous with respect to Lebesgue measure for almost all times, almost surely. The density possesses good regularity properties, and we obtain a moment bound for its (random) fractional Sobolev norm. This result is obtained for dynamics with a bounded drift, bounded and Hölder-continuous diffusion coefficients and when the diffusion coefficient for the idiosyncratic noise is uniformly elliptic. We directly study the sequence of particle systems via approximating their exchangeable dynamics by conditionally independent dynamics at the expense of an error. By using probabilistic means, the approximation and the error are controlled for each particle system, and the results are then passed to the large-system limit. The estimates thus obtained are combined via an analytic interpolation technique to derive a norm bound for the limiting random density. In this way, we obtain a regularity estimate for all cluster points, without requiring any knowledge of the dynamics of the limiting measure-valued flow.

math.PR↗

arXiv2025

A new approach to stochastic McKean-Vlasov limits with low-regularity coefficients

The empirical measure flow of a McKean-Vlasov $n$-particle system with common noise is a measure-valued process whose law solves an associated martingale problem. We obtain a stability result for the sequence of martingale problems: all narrow cluster points of the sequence of laws solve the formally limiting martingale problem. Through the solution of the limiting problem, we are able to characterize the dynamics of limits of the empirical measure flows. A major new aspect of our result is that it requires rather weak regularity assumptions for the coefficients, for instance a form of local continuity of the drift in the measure argument and ellipticity of the diffusion coefficient for the idiosyncratic noise. In fact, the formally limiting martingale problem may fail to have any solution if there are discontinuities in the measure argument, so that the stability property is in general not true under low regularity assumptions. Our novel approach leverages an emergence of regularity property for cluster points of the empirical measure flow. This provides us with a priori analytic regularity estimates which we use to compensate for the low regularity of the drift.

math.PR↗

arXiv2025

Existence for low-regularity McKean-Vlasov dynamics via emergence of regularity

We establish the existence of solutions to common noise McKean-Vlasov martingale problems for coefficients with low regularity. Our approach is able to handle the key challenge posed by drift coefficients that are discontinuous with respect to the narrow convergence of measures. This case arises for e.g. singular interactions. Our proof strategy proceeds via a two-step approximation using smoothed McKean-Vlasov $n$-particle systems: We first pass to the large system limit by taking $n\to \infty$, and subsequently remove the smoothing. A novel aspect of our work is the use of a crucial emergence of regularity property. It ensures that after the first limit, we obtain a process of measures that are absolutely continuous with respect to the Lebesgue measure and provides quantitative integrability bounds on their densities. We use this regularity to establish a tightness result in a stronger topology than is typically considered. In this way we obtain a sufficiently strong mode of convergence that lets us subsequently remove the smoothing and solve the McKean-Vlasov martingale problem via the particle system approximations.

math.PR↗

arXiv2019

The Tropical Division Problem and the Minkowski Factorization of Generalized Permutahedra

Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the Minkowski factorization of polytopes. Using our construction we show that for any given polytopal fan there is a polytope factorization basis, i.e. a finite set of polytopes with respect to which any polytope whose normal fan is refined by the original fan can be uniquely written as a signed Minkowski sum. We explicitly study the factorization of polymatroids and their generalizations, Coxeter matroid polytopes, and give a hyperplane description of the cone of deformations for this class of polytopes.

math.CO↗

arXiv2018

Tropical Geometry and Mechanism Design

We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry. We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set. These cells are the sets of incentive compatible payments and form tropical simplices, spanned by generating payments whose number equals the dimension of the simplex. The analysis of the collection of incentive compatible mechanisms via tropical simplices and their generating payments facilitates the use of geometric techniques. We use this view to derive a new geometric characterization of revenue equivalence but also show how to handle multidimensional mechanisms in the absence of revenue equivalence.

cs.GT↗