Scaling transformation of random walk distributions in a lattice

We use a decimation procedure in order to obtain the dynamical renormalization group transformation (RGT) properties of random walk distribution in a 1+1 lattice. We obtain an equation similar to the Chapman-Kolmogorov equation. First we show the existence of invariants through the RGT. We also show the existence of functions which are semi-invariants through the RGT. Second, we show as well that the distribution $R_q\mathrm{}\left(x\right)=[1+b(q-{1)x}^2{]}^{1/\left(1-q\right)}\hspace{0.5em}\mathrm{}(q>\mathrm{}1\mathrm{}),$ which is an exact solution of a nonlinear Fokker-Planck equation, is a semi-invariant for RGT. We obtain the map $q^{\prime }=f\left(q\right)\mathrm{}$ from the RGT and we show that this map has two fixed points: $q=1,$ attractor, and $q=2,$ repellor, which are the Gaussian and the Lorentzian, respectively. We show the connections between these result and the Levy flights.