The project aims to study certain key models involving non-local branching Markov processes and their connection with the corresponding nonlinear evolution equations, and specific stochastic methods. These models have a physical and social relevance, namely, the time evolution of a system of particles which move according to a Markov process, in which the occurrence of a fragmentation event (e.g., the splitting in « smaller particles ») and the interaction between the fragments are allowed, and branching processes from population dynamics. The related nonlinear PDEs and the Kolmogorov operators will play an essential role in the investigation. We use infinite dimensional stochastic analysis tools and non-local branching processes in studying stochastic representations of solutions of reaction-diffusion type equations. We are also concerned with other non-local and possibly non-Markovian processes like solutions to SDEs with Hermite and Rosemblatt noise, and Hunt processes associated to Mehler semigroups, whose construction and convergence to equilibria are going to be investigated. The results will be completed by probabilistic numerical aspects.