Open Access
Translator Disclaimer
January 2017 A criterion for convergence to super-Brownian motion on path space
Remco van der Hofstad, Mark Holmes, Edwin A. Perkins
Ann. Probab. 45(1): 278-376 (January 2017). DOI: 10.1214/14-AOP953


We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the $r$-point functions for $r=2,\ldots,5$. The $r$-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the $r$-point functions, but the lack of tightness has been an obstruction to proving convergence on path space.

We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.


Download Citation

Remco van der Hofstad. Mark Holmes. Edwin A. Perkins. "A criterion for convergence to super-Brownian motion on path space." Ann. Probab. 45 (1) 278 - 376, January 2017.


Received: 1 April 2014; Revised: 1 June 2014; Published: January 2017
First available in Project Euclid: 26 January 2017

zbMATH: 1364.82025
MathSciNet: MR3601651
Digital Object Identifier: 10.1214/14-AOP953

Primary: 60F17 , 60G68 , 60K35 , 82B41
Secondary: 05C05

Keywords: Functional limit theorem , Lattice trees , Super-Brownian motion

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 1 • January 2017
Back to Top