# Non-local Conservation Law from Stochastic Particle Systems

## TL;DR

This paper studies a system of interacting particles driven by Lévy processes and proves that, as the number of particles grows, their empirical density converges to a solution of a fractal conservation law, linking stochastic particle models to non-local PDEs.

## Contribution

It establishes a law of large numbers for the empirical density of Lévy-driven particle systems converging to a non-local conservation law in multiple dimensions.

## Key findings

- Empirical density converges uniformly to the PDE solution.
- The limiting PDE is a fractal conservation law.
- Results connect stochastic particle systems with non-local PDEs.

## Abstract

In this paper we consider an interacting particle system in $\mathbb{R}^d$ modelled as a system of $N$ stochastic differential equations driven by L\'evy processes. The limiting behaviour as the size $N$ grows to infinity is achieved as a law of large numbers for the empirical density process associated with the interacting particle system. We prove that the empirical process converges, uniformly in the space variable, to the solution of the $d$-dimensional fractal conservation law.