Abstract
For stochastic C0-semigroups on L1-spaces there is a wealth of results that show strong convergence to an equilibrium as t → ∞, given that the semigroup contains a partial integral operator. This has plenty of applications to transport equations and in mathematical biology. However, up to now partial integral operators do not play a prominent role in theorems which yield uniform convergence of the semigroup rather than only strong convergence.
In this article we prove that, for irreducible stochastic semigroups, uniform convergence to an equilibrium is actually equivalent to being partially integral and uniformly mean ergodic. In addition to this Tauberian theorem, we also show that our semigroup is uniformly convergent if and only if it is partially integral and the dual semigroup satisfies a certain irreducibility condition. Our proof is based on a uniform version of a lower bound theorem of Lasota and Yorke, which we combine with various techniques from Banach lattice theory.
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Acknowledgements
A special case of Theorem 1.1 was proved while the second-named author wrote his Master’s thesis at the Institute of Applied Analysis at Ulm University in winter 2017/18 [26].
The authors are indebted to Mustapha Mokhtar-Kharroubi for pointing out reference [22] to them.
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Glück, J., Martin, F.G. Uniform convergence of stochastic semigroups. Isr. J. Math. 247, 1–19 (2022). https://doi.org/10.1007/s11856-021-2240-z
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DOI: https://doi.org/10.1007/s11856-021-2240-z