  A general class of non-Markov, supercritical Gaussian branching particle
systems is introduced and its long-time asymptotics is studied. Both weak and
strong laws of large numbers are developed with the limit object being
characterized in terms of particle motion/mutation. Long memory processes, like
branching fractional Brownian motion and fractional Ornstein-Uhlenbeck
processes with large Hurst parameters, as well as rough processes, like
fractional processes with with smaller Hurst parameter, are included as
important examples. General branching with second moments is allowed and moment
measure techniques are utilized.
