In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.
Coleman, B.D., Gurtin, M.E.: Equipresence and constitutive equations for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2014)
Di Blasio, G., Kunisch, K., Sinestrari, E.: L2-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives. J. Math. Anal Appl. 102, 38–57 (1984)
Jeong, J.: Stabilizability of retarded functional differential equation in Hilbert space. Osaka J. Math. 28, 347–365 (1991)
Jeong, J., Nakagiri, S.I., Tanabe, H.: Structural operators and semigroups associated with functional differential equations in Hilbert spaces. Osaka J. Math. 30, 365–395 (1993)
Liu, K.: On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298 (2020)
Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–204 (1971)
Tanabe, H.: On fundamental solution of differential equation with time delay in Banach space. Proc. Jpn. Acad. 64, 131–180 (1988)
Tanabe, H.: Equations of Evolution Monographs and Studies in Mathematics, vol. 6. Pitman, London (1979)
This work is supported by Project of Tianjin Municipal Education Commission (Grant No. JW1714).
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Liu, K. Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay. Potential Anal (2021). https://doi.org/10.1007/s11118-021-09904-5
- Regularity property
- Fundamental solution
- Stochastic convolution
Mathematics Subject Classification (2010)