AN-ANNIHILATING–BRANCHING-PARTICLE-MODEL-FOR-THE-HEAT-EQUATION-WITH-AVERAGE-TEMPERATURE-ZERO.pdfVIP

AN ANNIHILATING – BRANCHING PARTICLE MODEL FOR THE HEAT EQUATION WITH AVERAGE TEMPERATURE ZERO Krzysztof Burdzy and Jeremy Quastel 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. 1. Introduction. A branching particle system representation for the heat equation solution with positive temperature was introduced in [BHIM] and later studied in [BHM] (see also [GK1] and [GK2]). Here is an informal description of that model and one of the main results, proved in [BHM]. Suppose that D is an open set in Rd and N Brownian particles move independently inside D. Whenever one of these particles hits the boundary of D, it is killed and one of the other particles, randomly chosen, splits into two particles, so that the number N of particles remains constant. When the number of particles goes to infinity and the initial (normalized) distribution of particles converges to a measure on D then the particle density Research partially supported by NSF grant DMS0303310 (KB) and NSERC (JQ). 1 converges to the solution of the heat equation for every time t ≥ 0, with the appropriate initial condition, n