\mu \rho_0$. Needless to say, both $\rho_0$ and $\tilde
\rho_0$ should lie between 0 and 1. Although the equivalence is shown only for the one-dimensional systems in Ref. [@PP05c], Eq.  is generally true for any $d$-dimensional systems, which can be easily shown by the same technique developed in Ref. [@PP05c].

  -- ------------------------------------------ -----------------
      $A\emptyset \leftrightarrow \emptyset A$  $D/d$
       $ A A \rightarrow \emptyset \emptyset$   $\lambda/d$
           $ A A \rightarrow A \emptyset$       $ {\eta}/(2d)$
           $ A A \rightarrow \emptyset A$       $ {\eta}/(2d) $
            $A   \rightarrow \emptyset$         $\xi$
           $\emptyset A \rightarrow A A $       ${\sigma}/(2d)$
          $ A \emptyset  \rightarrow A A $      $\sigma/(2d) $
  -- ------------------------------------------ -----------------

  : \[Table:DP\] $d$-dimensional reaction-diffusion processes of single species with hard core exclusion and their rates.

Now consider the branching annihilating random walks with one offspring (BAW1) [@TT92] which corresponds to the model with $\xi = 0$ in Table \[Table:DP\]. The parameters used in Ref. [@TT92] in one dimension are $D = p/2$, $\lambda = p$, and $\eta = \sigma = 1-p$ with the tuning parameter $p$. If $w \equiv
\mu - 1$ is very small and nonnegative, the solution of Eq.  up to the order of $w$ is $$\begin{aligned}
&\tilde D -D=  - \frac{w}{2} \sigma,\quad 