Scaling limit for a second-order particle system with local annihilation

TL;DR

This paper establishes a scaling limit for a second-order particle system with local annihilation, showing convergence to a degenerate elliptic PDE using advanced analytical techniques.

Contribution

It introduces a novel scaling limit for a second-order particle system with local annihilation, employing Green's function estimates and the Itô-Tanaka trick.

Findings

Empirical measure converges to a degenerate elliptic PDE.

Green's function estimates are crucial for the analysis.

The approach advances understanding of particle systems with annihilation.

Abstract

For a second-order particle system in $\mathbb R^d$ subject to locally-in-space pairwise annihilation, we prove a scaling limit for its empirical measure on position and velocity towards a degenerate elliptic partial differential equation. Crucial ingredients are Green's function estimates for the associated hypoelliptic operator and an It\^o-Tanaka trick.