Analysis and Probability Seminar

We study a branching annihilating random walk in which particles move on the d-dimensional lattice and evolve in discrete generations. Each particle produces a Poissonian number of offspring with mean μ which independently move to a uniformly chosen site within a fixed distance R from their parent’s position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. We prove that for any μ > 1 the process survives when R is sufficiently large. For fixed R we show that the process dies out if μ is too small or too large. Furthermore, for fixed (but large) R and 1 < μ < e^2 we exhibit an interval of μ-values for which the process survives. For such μ’s we can also show that the process has a unique non-trivial ergodic equilibrium and prove complete convergence starting from arbitrary initial conditions. The main techniques involve comparison with oriented percolation and coupling arguments.

Based on joint work with Alice Callegaro, Matthias Birkner, Jiřı́ Černý and Pascal Oswald