  Annihilation processes, where the reacting particles are influenced by some
external advective field, are one of the simplest examples of nonlinear
statistical systems. This type of processes can be observed in miscellaneous
chemical, biological or physical systems. In low space dimensions usual
description by means of kinetic rate equation is not sufficient and the effect
of density fluctuations must be taken into ac- count. Using perturbative
renormalization group we study the influ- ence of random velocity field on the
kinetics of single-species annihila- tion reaction at and below its critical
dimension $d_c = 2$. The advecting velocity field is modelled by the
self-similar in space Gaussian variable finite correlated in time
(Antonov-Kraichnan model). Effect of the compressibility of velocity field is
taken into account and the model is analyzed near its critical dimension by
means of three-parameter expansion in $\epsilon$, $\Delta$ and $\eta$. Here
$\epsilon$ is the deviation from the Kolmogorov scaling, $\Delta$ is the
deviation from the (critical) space dimension 2 and {\eta} is the deviation
from the parabolic dispersion law. Depending on the value of these exponents
and the value of compressiblity parameter {\alpha}, the studied model can
exhibit various asymptotic (long-time) regimes corresponding to the infrared
(IR) fixed points of the renormalization group. The possible regimes are
summarized and the decay rates for the mean particle number are calculated in
the leading order of the perturbation theory.
