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.
"An annihilating–branching particle model for the heat equation with average temperature zero." Ann. Probab. 34 (6) 2382 - 2405, November 2006. https://doi.org/10.1214/009117906000000511