The Annals of Probability

An annihilating–branching particle model for the heat equation with average temperature zero

Krzysztof Burdzy and Jeremy Quastel

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Abstract

We consider two species of particles performing random walks in a domain in ℝd with reflecting boundary conditions, which annihilate on contact. In addition, there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species annihilate each other, particles of each species, chosen at random, give birth. We assume initially equal numbers of each species and show that the system has a diffusive scaling limit in which the densities of the two species are well approximated by the positive and negative parts of the solution of the heat equation normalized to have constant L1 norm. In particular, the higher Neumann eigenfunctions appear as asymptotically stable states at the diffusive time scale.

Article information

Source
Ann. Probab., Volume 34, Number 6 (2006), 2382-2405.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1171377448

Digital Object Identifier
doi:10.1214/009117906000000511

Mathematical Reviews number (MathSciNet)
MR2294987

Zentralblatt MATH identifier
1122.60085

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Branching random walk Neumann eigenfunction heat equation hydrodynamic limit

Citation

Burdzy, Krzysztof; Quastel, Jeremy. An annihilating–branching particle model for the heat equation with average temperature zero. Ann. Probab. 34 (2006), no. 6, 2382--2405. doi:10.1214/009117906000000511. https://projecteuclid.org/euclid.aop/1171377448


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