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.
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References
Butzer, P., Berens, H.: Semi-Groups of Operators and Approximation. Springer, New York (1967)
Chojnowska-Michalik, A.: Representation theorem for general stochastic delay equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. 26, 634–641 (1978)
Chojnowska-Michalik, A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stochastics 21, 251–287 (1987)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Da Prato, G., Kwapien, S., Zabczyk, J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23, 1–23 (1987)
Di Blasio, G., Kunisch, K., Sinestrari, E.: L 2-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives. J. Math. Anal. Appl. 102, 38–57 (1984)
Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)
Lions, J., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. I. Springer, Berlin (1972)
Liu, K.: Stochastic retarded evolution equations: Green operators, convolutions and solutions. Stoch. Anal. Appl. 26, 624–650 (2008)
Liu, K.: Stationary solutions of retarded Ornstein–Uhlenbeck processes in Hilbert spaces. Stat. Probab. Lett. 78, 1775–1783 (2008)
Liu, K.: The fundamental solution and its role in the optimal control of infinite dimensional neutral systems. Appl. Math. Optim. 60, 1–38 (2009)
Liu, K.: Retarded stationary Ornstein–Uhlenbeck processes driven by Lévy noise and operator self-decomposability. Potential Anal. 33, 291–312 (2010)
Liu, K.: A criterion for stationary solutions of retarded linear equations with additive noise. Stoch. Anal. Appl. 29 (2011, to appear)
Lunardi, A.: On the evolution operator for abstract parabolic equations. Isr. J. Math. 60, 281–314 (1987)
Nakagiri, S.: Optimal control of linear retarded systems in Banach spaces. J. Math. Anal. Appl. 120, 169–210 (1986)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Tanabe, H.: Fundamental solutions for linear retarded functional differential equations in Banach space. Funkc. Ekvacioj 35, 149–177 (1992)
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Liu, K. On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces. J Theor Probab 25, 565–593 (2012). https://doi.org/10.1007/s10959-011-0374-3
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DOI: https://doi.org/10.1007/s10959-011-0374-3