The structure of the population inside the propagating front

We investigate the long-time and long-range behavior of branching diffusion processes (starting with a single particle) in two different situations: when the branching potential has compact support and when it is constant. For compactly supported branching potentials, the qualitative behavior of the processes depends on the intensity of the branching. We analyze the super-critical case, when the total number of particles growing exponentially with positive probability. We study the asymptotics of the number of particles in different regions of space and describe the growth of the region occupied by the particles. For a positive constant branching potential, we observe the effect of intermittency: for the number of particles located at time t in a unit neighborhood of a point tv, the k-th moment grows exponentially faster than the k-th power of the first moment as t → ∞, provided that v ̸= 0 and k = k(v) is sufficiently large. This effect is not present in the case of a compactly supported branching potential. 2000 Mathematics Subject Classification Numbers: 60J80, 35K10.