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April  2020, 25(4): 1279-1298. doi: 10.3934/dcdsb.2019220

## On regularity of stochastic convolutions of functional linear differential equations with memory

 a. School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China b. Department of Mathematical Sciences, School of Physical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

The author is grateful to the Tianjin Thousand Talents Plan for its financial support

Received  April 2019 Published  April 2020 Early access  September 2019

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $C_0$-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

Citation: Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220
##### References:
 [1] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912. [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Second Edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513. [3] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  doi: 10.1007/BF02761404. [4] J. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math., 28 (1991), 347-365. [5] J. Jeong, S. I. Nakagiri and H. Tanabe, Structural operators and semigroups associated with functional differential equations in Hilbert spaces, Osaka J. Math., 30 (1993), 365-395. [6] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683. [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, , Applied Mathematical Sciences, Vol. 44. Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [8] J. Prüss, On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations, 5 (1983), 211-236. [9] E. Sinestrari, On a class of retarded partial differential equations of Volterra type, Math. Z., 186 (1984), 223-246.  doi: 10.1007/BF01161806. [10] E. Sinestrari, A noncompact differentiable semigroup arising from an abstract delay equation, C. R. Math. Rep. Acad. Sci. Canada., 6 (1984), 43-48. [11] H. Tanabe, On fundamental solution of differential equation with time delay in Banach space, Proc. Japan Acad., 64 (1988), 131-134.  doi: 10.3792/pjaa.64.131. [12] H. Tanabe, Fundamental solutions for linear retarded functional differential equations in Banach spaces, Funkcialaj Ekvacioj, 35 (1992), 149-177.

show all references

##### References:
 [1] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912. [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Second Edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513. [3] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  doi: 10.1007/BF02761404. [4] J. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math., 28 (1991), 347-365. [5] J. Jeong, S. I. Nakagiri and H. Tanabe, Structural operators and semigroups associated with functional differential equations in Hilbert spaces, Osaka J. Math., 30 (1993), 365-395. [6] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683. [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, , Applied Mathematical Sciences, Vol. 44. Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [8] J. Prüss, On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations, 5 (1983), 211-236. [9] E. Sinestrari, On a class of retarded partial differential equations of Volterra type, Math. Z., 186 (1984), 223-246.  doi: 10.1007/BF01161806. [10] E. Sinestrari, A noncompact differentiable semigroup arising from an abstract delay equation, C. R. Math. Rep. Acad. Sci. Canada., 6 (1984), 43-48. [11] H. Tanabe, On fundamental solution of differential equation with time delay in Banach space, Proc. Japan Acad., 64 (1988), 131-134.  doi: 10.3792/pjaa.64.131. [12] H. Tanabe, Fundamental solutions for linear retarded functional differential equations in Banach spaces, Funkcialaj Ekvacioj, 35 (1992), 149-177.
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