Stochastic basins of attraction and generalized committor functions

We study two generalizations of the basin of attraction of a stable state, to the case of stochastic dynamics, arbitrary regions, and finite-time horizons. This is done by introducing generalized committor functions and studying soujourn times. We show that the volume of the generalized basin, the basin stability, can be efficiently estimated using Monte Carlo-like techniques, making this concept amenable to the study of high-dimension stochastic systems. Finally, we illustrate in a set of examples that stochastic basins efficiently capture the realm of attraction of metastable sets, which parts of phase space go into long transients in deterministic systems, that they allow us to deal with numerical noise, and can detect the collapse of metastability in high-dimensional systems. We discuss two far-reaching generalizations of the basin of attraction of an attractor. The basin of attraction of an attractor are those states that eventually will get to the attractor. In a generic stochastic system, all regions will be left again; no attraction is permanent. To obtain the equivalent of the basin of attraction of a region we need to generalize the notion to cover finite-time horizons and finite regions. We do so by considering soujourn times, the fraction of time that a trajectory spends in a set, and by generalizing committor functions which arise in the study of hitting probabilities. In a simplified setting we show that these two notions reduce to the normal notions of the basin of attraction in the appropriate limits. We also show that the volume of these stochastic basins can be efficiently estimated for high-dimensional systems at computational cost comparable to that for deterministic systems. To fully illustrate the properties captured by the stochastic basins, we show a set of examples ranging from simple conceptual models to high-dimensional inhomogeneous oscillator chains. These show that stochastic basins efficiently capture metastable attraction, the presence of long transients, that they allow us to deal with numerical and approximation noise, and can detect the collapse of metastability with increasing noise in high-dimensional systems.