The heat equation with order-respecting absorption and particle systems with topological interaction

Rami Atar

Published 2020-11-15Version 1

A PDE formulation is proposed, referred to as a heat equation with order-respecting absorption, aimed at characterizing hydrodynamic limits of a class of particle systems on the line with topological interaction that have so far been described by free boundary problems. It consists of the heat equation with measure-valued injection and absorption terms, where the absorption measure respects the usual order on $\R$ in the sense that, for all $r\in\R$, it charges $(-\iy,r)$ only at times when the solution vanishes on $(r,\iy)$. The formulation is used to obtain new hydrodynamic limit results for two models. One is a variant of the main model studied by Carinci, De Masi, Giardin\`a and Presutti \cite{CDGPbook} where Brownian particles undergo injection according to a general injection measure, and removal that is restricted to the rightmost particle of the configuration. This partially addresses a conjecture of \cite{CDGPbook}. Next a Brownian particle system is considered where the $Q$-quantile member of the population is removed until extinction, where $Q$ is a given $[0,1]$-valued continuous function of time. Here, unlike in earlier work on the subject, the removal mechanism acts on particles that are `at the boundary' but are not rightmost or leftmost. Finally, further potential uses of order-respecting absorption are mentioned.