This paper studies systems of particles following independent random walks and subject to annihilation, binary branching, coalescence, and deaths. In the case without annihilation, such systems have been studied in our 2005 paper "Branching-coalescing particle systems". The case with annihilation is considerably more difficult, mainly as a consequence of the non monotonicity of such systems and a more complicated duality. Nevertheless, we show that adding annihilation does not significantly change the long-time behavior of the process and in fact, systems with annihilation can be obtained by thinning systems without annihilation.
"Systems of branching, annihilating, and coalescing particles." Electron. J. Probab. 17 1 - 32, 2012. https://doi.org/10.1214/EJP.v17-2003