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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 smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment measure techniques are utilized. [ABSTRACT FROM AUTHOR]

This paper studies the weak and strong solutions to the stochastic differential equation d X ( t ) = − 1 2 W ̇ ( X ( t ) ) d t + d B ( t ) , where ( B ( t ) , t ≥ 0 ) is a standard Brownian motion and W ( x ) is a two sided Brownian motion, independent of B . It is shown that the Itô–McKean representation associated with any Brownian motion (independent of W ) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ∫ 0 T W ̇ ( X ( t ) ) d t , the main idea is to use the concept of local time together with the polygonal approximation W π . Some new results on the local time of Brownian motion needed in our proof are established. [ABSTRACT FROM AUTHOR]