Abstract
The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.
Highlights
Let (Ω, (Ft)t≥0, (Xt)t≥0, (Px)x∈E∪{∂}) be a time homogeneous Markov process with state space E ∪ {∂}, where E is a measurable space
We assume that ∂ ∈ E is an absorbing state for the process, which means that Xs = ∂ implies Xt = ∂ for all t ≥ s, Px-almost surely for all x ∈ E
It is well known that a probability measure α is a quasi-stationary distribution if and only if there exists a probability measure μ on E such that lim Pμ(Xt ∈ A | t < τ∂) = α(A)
Summary
Let (Ω, (Ft)t≥0, (Xt)t≥0, (Px)x∈E∪{∂}) be a time homogeneous Markov process with state space E ∪ {∂}, where E is a measurable space. The first condition implies that, in cases of unbounded state space E (like N or R+), the process (Xt, t ≥ 0) comes down from infinity in the sense that, there exists a compact set K ⊂ E such that infx∈E Px(Xt0 ∈ K | t0 < τ∂ ) > 0 This property is standard for biological population processes such as Lotka-Volterra birth and death or diffusion processes [1, 3]. Is well defined and the process (Ω, (Ft)t≥0, (Xt)t≥0, (Qx)x∈E) is an E-valued homogeneous Markov process This process admits the unique invariant probability measure (sometimes refered to as the doubly limiting quasi-stationary distribution [5]). The second aim of this note is to prove that the existence of the Q-process with uniform bounds in (1.4) and its uniform exponential ergodicity (1.5) form a necessary and sufficient condition for the uniform exponential convergence (1.2) toward a unique quasi-stationary distribution
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