On the extinction of Dickman's reaction- diffusion processes

Some comments on the extinction of processes are given for the reaction-diffusion processes recently proposed by Dickman as mathematical models for chemical reactions on catalytic surfaces. A brief review of mean-field-type approximations (MFA) is presented for three models; the single annihilation model (SAM), the pair annihilation model (PAM) and the triplet annihilation model (TAM). Two theorems on the extinction are proved. The former supports the MFA predictions for the SAM. The latter gives a qualitative correction to the phase diagram obtained by the MFA for the PAM in low dimensions (d ⩽ 2). In order to obtain the latter theorem, we discuss the relationship between the PAM and the branching annihilating random walk of Bramson and Gray.