Symmetric cooperative motion in one dimension

TL;DR

This paper studies symmetric cooperative motions, a class of random processes, establishing their distributional convergence and linking their behavior to solutions of certain one-dimensional PDEs, including a new proof of the Bernoulli CLT.

Contribution

It introduces symmetric cooperative motions, proves their distributional convergence, and connects these processes to solutions of the porous medium and p-Laplace equations in one dimension.

Findings

Distributional convergence of symmetric cooperative motions

New proof of the Bernoulli central limit theorem

Relation between distributional and viscosity solutions of PDEs

Abstract

We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, arXiv:1909.07367]. We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addition, we prove a PDE result relating distributional solutions and viscosity solutions of the porous medium equation and the parabolic $p$-Laplace equation, respectively, in one dimension.