Uniform convergence of conditional distributions for one-dimensional diffusion processes

In this paper, we study the quasi-stationary behavior of the one-dimensional diffusion process with a regular or exit boundary at 0 and an entrance boundary at $\infty$. By using the Doob's $h$-transform, we show that the conditional distribution of the process converges to its unique quasi-stationary distribution exponentially fast in the total variation norm, uniformly with respect to the initial distribution. Moreover, we also use the same method to show that the conditional distribution of the process converges exponentially fast in the $\psi$-norm to the unique quasi-stationary distribution.Comment: 15 page

Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing

This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusion processes with killing on $[0,\infty)$. We obtain criteria for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. Our approach is based on probabilistic and coupling methods, contrary to the classical approach based on spectral theory results. Our general criteria apply in the case where $\infty$ is entrance and 0 either regular or exit, and are proved to be satisfied under several explicit assumptions expressed only in terms of the speed and killing measures. We also obtain exponential ergodicity results on the $Q$-process. We provide several examples and extensions, including diffusions with singular speed and killing measures, general models of population dynamics, drifted Brownian motions and some one-dimensional processes with jumps.Comment: arXiv admin note: text overlap with arXiv:1506.0238

Convergence to non-minimal quasi-stationary distributions for one-dimensional diffusions and its application to Kummer diffusions (Probability Symposium)

We summarize the following results of the author's recent work [17] without proof. For one-dimensional diffusions killed at the boundaries, the domain of attraction of non-minimal quasi-stationary distributions is studied. We give a general method of reducing the convergence to the tail behavior of the lifetime via a property which we call the first hitting uniqueness. We apply the result to Kummer diffusions with negative drifts and clarify the domain of attraction of each non-minimal quasi-stationary distribution for the processes

Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates

This article studies the quasi-stationary behaviour of multidimensional birth and death processes, modeling the interaction between several species, absorbed when one of the coordinates hits 0. We study models where the absorption rate is not uniformly bounded, contrary to most of the previous works. To handle this natural situation, we develop original Lyapunov function arguments that might apply in other situations with unbounded killing rates. We obtain the exponential convergence in total variation of the conditional distributions to a unique stationary distribution, uniformly with respect to the initial distribution. Our results cover general birth and death models with stronger intra-specific than inter-specific competition, and cases with neutral competition with explicit conditions on the dimension of the process.Comment: 18 page