Stochastic Stability of the Classical Lorenz Flow Under Impulsive Type Forcing

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in Keller (Attractors and bifurcations of the stochastic Lorenz system Report 389, Institut für Dynamische Systeme, Universität Bremen, 1996), Kifer (Random Perturbations of Dynamical Systems. Birkhäuser, Basel, 1988), and Metzger (Commun. Math. Phys. 212, 277–296, 2000). Namely, given a cross-sectionMfor the unperturbed flow, each time the trajectory of the system crossesMthe phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the umperturbed flow is then carried on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes.