Harmonic analysis of random walks with heavy tails

We consider a continuous-time symmetric, spatially homogeneous branching random walk on a multidimensional lattice with a single branching source. Corresponding transition intensities of the underlying random walk are assumed to have heavy tails. This assumption implies that the variance of jumps is infinite. The growth rate estimate of the Fourier transform for transition intensities and of the asymptotics of the mean number of particles in the source in subcritical case are obtained.