Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times

TL;DR

This work analyzes a system of reflecting Brownian particles on $[0,\infty)$ interacting through local times, showing that strong feedback can cause finite-time breakdown and characterizing this breakdown via spectral data of active-node matrices. It develops both finite-particle dynamics and a symmetric mean-field limit, proving existence, uniqueness, and propagation of chaos for the McKean--Vlasov SDE with reflection, and connecting these stochastic dynamics to a nonlinear Fokker–Planck equation. The paper further identifies regimes (subcritical, critical, supercritical) with distinct long-time behaviors, including explicit stationary states at $\alpha=1$ and self-similar profiles for $\alpha<1$, and provides a probabilistic interpretation as a liquidity-sharing model in finance and a link to the supercooled Stefan problem. These results rigorously bridge stochastic interacting particle systems with boundary interactions and macroscopic PDE descriptions, offering insights into phase transitions, breakdown phenomena, and potential applications in finance and cell polarization.

Abstract

We study a system of reflecting Brownian motions on the positive half-line in which each particle has a drift toward the origin determined by the local times at the origin of all the particles. We show that if this local time drift is too strong, such systems can exhibit a breakdown in their solutions in that there is a time beyond which the system cannot be extended. In the finite particle case we give a complete characterization of the finite time breakdown, relying on a novel dynamic graph structure. We consider the mean-field limit of the system in the symmetric case and show that there is a McKean--Vlasov representation. If the drift is too strong, the solution to the corresponding Fokker--Planck equation has a blow up in its solution. We also establish the existence of stationary and self-similar solutions to the McKean--Vlasov equation in the case where there is no breakdown of the system. This work is motivated by models for liquidity in financial markets, the supercooled Stefan problem, and a toy model for cell polarization.

Figures (2)

• Figure 1: The plot shows trajectories of a particle approximation ($N = 10^5$) of the unique maximal solution $(\ell, T)$ of McKean--Vlasov SDE \ref{['eq:non_linear_skorokhod']} for different interactions strength $\alpha = n/4$ with $n \in \{0, \dots, 8\}$. If $n \in \{5, \dots, 8\}$, so $\alpha > 1$, the system breaks down before time $1$.
• Figure 2: The plot on the left-hand side shows the true stationary profile for $\alpha = 1$ as well as an approximation of the density of $X_t$ for $t = 10$ for the unique solution of McKean--Vlasov SDE \ref{['eq:non_linear_skorokhod']} with $\alpha = 1$ based on the finite system \ref{['eq:ps']} with $N = 10^5$ particles. The right panel shows the true self-similar profile for $\alpha = 1/2$ as well as an approximation of the density of $X_t/\mathbb{E} X_t$ based on the same particle system with $\alpha = 1/2$.

Theorems & Definitions (65)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Definition 1.8
• Remark 1.9
• Lemma 1.10