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On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces

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  • Kai Liu

    (The University of Liverpool)

Abstract

In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite-dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of the Burkholder–Davis–Gundy inequality for stochastic convolutions.

Suggested Citation

  • Kai Liu, 2012. "On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces," Journal of Theoretical Probability, Springer, vol. 25(2), pages 565-593, June.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:2:d:10.1007_s10959-011-0374-3
    DOI: 10.1007/s10959-011-0374-3
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    References listed on IDEAS

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    1. Liu, Kai, 2008. "Stationary solutions of retarded Ornstein-Uhlenbeck processes in Hilbert spaces," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1775-1783, September.
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    Cited by:

    1. Seyedeh Marzieh Ghavidel, 2023. "A characterization of well‐posedness for the second order abstract Cauchy problems with finite delay," Mathematische Nachrichten, Wiley Blackwell, vol. 296(6), pages 2352-2365, June.

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