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**An annihilating-branching particle model for the heat equation with average temperature zero.**
*(English)*
Zbl 1122.60085

The authors consider two species of particles performing random walks in a domain in \(\mathbb R^d\) with reflecting boundary conditions which annihilate on contact. When two particles of different species annihilate each other, particles of each species, chosen at random, produce children. Thus, the total number of particles of each type is preserved.

Assuming that initially the population consists of equal numbers of each species, the authors show that the system has a diffusive scaling limit in which the densities of the two species are approximated by the positive and negative of the solution of a heat equation. The proof is based on an observation that the microscopic evolution of the density \(\eta\) of the difference between the occupation numbers of the two types of particles has the form \(d\eta (t)=(\Delta +V(t))\eta(t) dt+dM(t)\) where \(M(t)\) is a martingale, \(\Delta\) is the discrete Laplace operator, and \(V(t)\) is the rate of annihilation.

Assuming that initially the population consists of equal numbers of each species, the authors show that the system has a diffusive scaling limit in which the densities of the two species are approximated by the positive and negative of the solution of a heat equation. The proof is based on an observation that the microscopic evolution of the density \(\eta\) of the difference between the occupation numbers of the two types of particles has the form \(d\eta (t)=(\Delta +V(t))\eta(t) dt+dM(t)\) where \(M(t)\) is a martingale, \(\Delta\) is the discrete Laplace operator, and \(V(t)\) is the rate of annihilation.

Reviewer: Vladimir Vatutin (Moskva)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60F17 | Functional limit theorems; invariance principles |

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\textit{K. Burdzy} and \textit{J. Quastel}, Ann. Probab. 34, No. 6, 2382--2405 (2006; Zbl 1122.60085)

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