Abstract
This paper improves on the theory part of Williams [12]1 in that results assumed there are here proved. The Wiener-Hopf factorization is now formulated in terms of non-negative operators, so that problems concerning domains of infinitesimal generators and the like do not arise.
Duality results are proved for’ symmetric’ cases by extending two important probabilistic ideas for the Markov-chain context from Kennedy [4] and finding a way to transfer the chain arguments to general Markov processes. The whole functional-analysis picture for ‘symmetric’ cases becomes much more illuminating under the assumption that a certain strict-contraction property holds. Then Hilbertspace spectral theory meshes extraordinarily well with Probability Theory. Even for a 2×2 Markov chain, the strict-contraction property may fail. On the other hand. Williams and Marles [14] showed that the property holds in a wide variety of important cases including some where it might be expected to fail. It is hoped to throw more light on the property in a subsequent paper.
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References
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Williams, D. (2008). A new look at ‘Markovian’ Wiener-Hopf theory. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_16
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