On the moments of branching random walk in a random medium with a Gumbelian potential

A time-continuous branching random walk over a multidimensional lattice in a random medium is considered. Underlying random walk is considered to be simple and symmetric. The random medium at each point of the lattice is determined by non-negative, independent, and equally distributed random intensities of splitting and death of particles. It is assumed that the difference in the intensities of splitting and death of particles has an asymptotically Gumbelian distribution. The limiting behavior of the moments averaged over the medium is obtained.