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**Multiscale couplings in prototype hybrid deterministic/stochastic systems. I: Deterministic closures.**
*(English)*
Zbl 1103.93013

Summary: We introduce and study a class of model prototype hybrid systems comprised of a microscopic stochastic surface process modeling adsorption/desorption and/or surface diffusion of particles coupled to an ordinary differential equation (ODE) displaying bifurcations excited by a critical noise parameter. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic models.

We obtain deterministic mesoscopic models from the hybrid system by employing two methods: stochastic averaging principle and mean field closures. In this paper we focus on the case where phase transitions do not occur in the stochastic system. In the averaging principle case a faster stochastic mechanism is assumed compared to the ODE relaxation and a local equilibrium is induced with respect to the Gibbs measure on the lattice system. Under these circumstances remarkable agreement is observed between the hybrid system and the averaged system predictions. We exhibit several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory. As expected fluctuation driven rare events do occur in several parameter regimes which could not possibly be captured by the deterministic averaging principle equation.

We obtain deterministic mesoscopic models from the hybrid system by employing two methods: stochastic averaging principle and mean field closures. In this paper we focus on the case where phase transitions do not occur in the stochastic system. In the averaging principle case a faster stochastic mechanism is assumed compared to the ODE relaxation and a local equilibrium is induced with respect to the Gibbs measure on the lattice system. Under these circumstances remarkable agreement is observed between the hybrid system and the averaged system predictions. We exhibit several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory. As expected fluctuation driven rare events do occur in several parameter regimes which could not possibly be captured by the deterministic averaging principle equation.

### MSC:

93A30 | Mathematical modelling of systems (MSC2010) |

93C65 | Discrete event control/observation systems |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

93E03 | Stochastic systems in control theory (general) |