Abstract
The goal of this note is to show how recent results on the theory of quasi-stationary distributions allow to deduce effortlessly general criteria for the geometric convergence of normalized unbounded semigroups.
Highlights
Let E be a measurable space and (Pn, n ∈ Z+) be a positive semigroup of operators on the space L∞(ψ1) to itself, where ψ1 : E → (0, +∞) is measurable and L∞(ψ1) is the set of measurable f : E →R such that | f |/ψ1 is bounded, endowed with the norm f ψ1 = | f |/ψ1 ∞
The asymptotic behavior of such semigroups is the subject of the theory of quasi-stationary distributions based on various tools, including the theory of R-recurrent Markov chains [31, 29, 28, 17], spectral theoretic results
We show here how these R-positivity criteria can be directly derived as corollaries of the results of [10], applied to the sub-Markov semigroup (Qn, n ∈ Z+)
Summary
Let E be a measurable space and (Pn, n ∈ Z+) be a positive semigroup of operators on the space L∞(ψ1) to itself, where ψ1 : E → (0, +∞) is measurable and L∞(ψ1) is the set of measurable f : E →. Our results provide practical criteria for the general theory of R-positive recurrence of unbounded semigroups as developed in [29, Section 6.2] and [28]. We show here how these R-positivity criteria can be directly derived as corollaries of the results of [10], applied to the sub-Markov semigroup (Qn, n ∈ Z+). This approach has the advantage to allow one to deduce with little extra effort sufficient criteria for the convergence of unbounded semigroups from the abundant theory of sub-Markov processes
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