Study of anomalous kinetics of the annihilation reaction $A+A\to\varnothing$

Using the perturbative renormalization group, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction $A+A\to\varnothing$ at and below its critical dimension $d_c=2$. We use the second-quantization formalism of Doi to bring the stochastic problem to a field theory form. We investigate the reaction in spaces of dimension $d\sim2$ using a two-parameter expansion in $\epsilon$ and $\Delta$, where $\epsilon$ is the deviation from the Kolmogorov scaling parameter and $\Delta$ is the deviation from the space dimension $d=2$. We evaluate all the necessary quantities, including fixed points with their regions of stability, up to the second order of the perturbation theory.