Asymptotic upper bound of density for two-particle annihilating exclusion

We consider a stochastic process which presents an evolution of particles of two types on ^d with annihilations between particles of opposite types. Initially, at each site of ^d, independently of the other sites, we put a particle with probability 21 and assign to it one of two types with equal chances. Each particle, independently from the others, waits an exponential time with mean 1, chooses one of its neighboring sites on the lattice ^d with equal probabilities, and jumps to the site chosen. If the site to which a particle attempts to move is occupied by another particle of the same type, the jump is suppressed; if it is occupied by a particle of the opposite type, then both are annihilated and disappear from the system. The considered process may serve as a model for the chemical reaction A+B inert. The paper concerns an upper bound of(t), the density of particles in the system at timet. We prove that(t)<t-d/ whent>t() for all>0 in the dimensionsd4 and asymptotically(t)^–1 in the higher dimensions. In our proofs, we used the ideas and the technique developed by Bramson and Lebowitz and the tools which are customarily used to study a symmetric exclusion process.