In this paper we study quasi-stationarity for a large class of Kolmogorov
diffusions. The main novelty here is that we allow the drift to go to−∞at the
origin, and the diffusion to have an entrance boundary at +∞. These diffusions
arise as images, by a deterministic map, of generalized Feller diffusions,
which themselves are obtained as limits of rescaled birth–death processes.
Generalized Feller diffusions take nonnegative values and are absorbed at
zero in finite time with probability 1. An important example is the logistic
Feller diffusion.
We give sufficient conditions on the drift near 0 and near +∞ for the existence
of quasi-stationary distributions, as well as rate of convergence in the
Yaglom limit and existence of the Q-process. We also show that, under these
conditions, there is exactly one quasi-stationary distribution, and that this distribution
attracts all initial distributions under the conditional evolution, if
and only if +∞ is an entrance boundary. In particular, this gives a sufficient
condition for the uniqueness of quasi-stationary distributions. In the proofs
spectral theory plays an important role on L2 of the reference measure for
the killed process
For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non uniformly hyperbolic dynamical systems developed by L.S.Young.
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