Criteria for convergence to super-Brownian motion on path space

R.W. Hofstad, van der, M.P. Holmes, E.A. Perkins

Onderzoeksoutput: Boek/rapportRapportAcademic

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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, …, 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 criteria will also apply to the other statistical physics models cited above.
Originele taal-2Engels
Plaats van productieEindhoven
UitgeverijEurandom
Aantal pagina's68
StatusGepubliceerd - 2014

Publicatie series

NaamReport Eurandom
Volume2014006
ISSN van geprinte versie1389-2355

Vingerafdruk

Super-Brownian Motion
Path Space
Tightness
Statistical Physics
Oriented Percolation
Measure-valued Process
Branching Random Walk
Contact Process
Spatial Structure
Obstruction
High-dimensional
Converge
Sufficient Conditions
Model

Citeer dit

Hofstad, van der, R. W., Holmes, M. P., & Perkins, E. A. (2014). Criteria for convergence to super-Brownian motion on path space. (Report Eurandom; Vol. 2014006). Eindhoven: Eurandom.
Hofstad, van der, R.W. ; Holmes, M.P. ; Perkins, E.A. / Criteria for convergence to super-Brownian motion on path space. Eindhoven : Eurandom, 2014. 68 blz. (Report Eurandom).
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abstract = "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, …, 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 criteria will also apply to the other statistical physics models cited above.",
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Hofstad, van der, RW, Holmes, MP & Perkins, EA 2014, Criteria for convergence to super-Brownian motion on path space. Report Eurandom, vol. 2014006, Eurandom, Eindhoven.

Criteria for convergence to super-Brownian motion on path space. / Hofstad, van der, R.W.; Holmes, M.P.; Perkins, E.A.

Eindhoven : Eurandom, 2014. 68 blz. (Report Eurandom; Vol. 2014006).

Onderzoeksoutput: Boek/rapportRapportAcademic

TY - BOOK

T1 - Criteria for convergence to super-Brownian motion on path space

AU - Hofstad, van der, R.W.

AU - Holmes, M.P.

AU - Perkins, E.A.

PY - 2014

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N2 - 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, …, 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 criteria will also apply to the other statistical physics models cited above.

AB - 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, …, 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 criteria will also apply to the other statistical physics models cited above.

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Hofstad, van der RW, Holmes MP, Perkins EA. Criteria for convergence to super-Brownian motion on path space. Eindhoven: Eurandom, 2014. 68 blz. (Report Eurandom).