SPDE Approximation for Random Trees

TL;DR

This paper studies the genealogy of critical branching processes conditioned on survival, showing that rescaled ancestor-descendant relations converge to a continuum tree described by an SPDE driven by a Brownian sheet.

Contribution

It introduces a novel SPDE framework for approximating the genealogy of critical branching processes via a continuum tree limit.

Findings

Convergence of rescaled monotone maps to a limiting flow

The limiting flow solves an SPDE with a Brownian sheet

Establishment of a continuum tree model for critical branching processes

Abstract

We consider the genealogy tree for a critical branching process conditioned on non-extinction. We enumerate vertices in each generation of the tree so that for each two generations one can define a monotone map describing the ancestor--descendant relation between their vertices. We show that under appropriate rescaling this family of monotone maps converges in distribution in a special topology to a limiting flow of discontinuous monotone maps which can be seen as a continuum tree. This flow is a solution of an SPDE with respect to a Brownian sheet.