June  2021, 26(6): 3235-3269. doi: 10.3934/dcdsb.2020226

Invariant measures of stochastic delay lattice systems

1. 

School of Mathematics, Shandong University, Jinan 250100, China

2. 

School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005, China

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Bixiang Wang

Received  January 2020 Revised  May 2020 Published  June 2021 Early access  July 2020

Fund Project: Zhang Chen is partially supported by NNSF of China grant 11471190, 11971260, NSF of Shandong Province grant ZR2014AM002. Xiliang Li is partially supported by NNSF of China grant 11971273 and NSF of Shandong Province grant ZR2018MA004

This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small.

Citation: Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226
References:
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V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631-637.  doi: 10.1142/S0218127494000459.

[2]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[3]

P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.

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P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, International J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[5]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[7]

J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501.

[8]

W. J. Beyn and S. Y. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 15 (2003), 485-515.  doi: 10.1023/B:JODY.0000009745.41889.30.

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Z. BrzezniakM. Ondrejat and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differential Equations, 260 (2016), 4157-4179.  doi: 10.1016/j.jde.2015.11.007.

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Z. BrzezniakE. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Annals of Probability, 45 (2017), 3145-3201.  doi: 10.1214/16-AOP1133.

[11]

O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), 1-23.  doi: 10.1214/17-EJP122.

[12]

T. CaraballoM. J. Garrido-Atienza and B. Schmalfuss, Exponential stability of stationary solutions for semilinear stochastic evolution equations with delays, Discret. Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.

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T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[14]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[15]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

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J. Eckmann and M. Hairer, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, 14 (2001), 133-151.  doi: 10.1088/0951-7715/14/1/308.

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C. E. Elmer and E. S. Van Vleck, Analysis and computation of traveling wave solutions of bistable differential-difference equations, Nonlinearity, 12 (1999), 771-798.  doi: 10.1088/0951-7715/12/4/303.

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A. Es-SarhirM. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200. 

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M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stochastics and Dynamics, 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.

[29]

K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[30]

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J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[32]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.

[33]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[34]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.

[35]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.

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Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

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K. Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.

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R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.

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N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrodinger equation, J. Differential Equations, 217 (2005), 88-123.  doi: 10.1016/j.jde.2005.06.002.

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J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.

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J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.

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J. Kim, On the stochastic Benjamin-Ono equation, J. Differential Equations, 228 (2006), 737-768.  doi: 10.1016/j.jde.2005.11.005.

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J. Kim, Periodic and invariant measures for stochastic wave equations, Electronic Journal of Differential Equations, 2004 (2004), 1-30. 

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J. Kim, Invariant measures for a stochastic nonlinear Schrodinger equation, Indiana University Mathematics Journal, 55 (2006), 687-717.  doi: 10.1512/iumj.2006.55.2701.

[46] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986. 
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Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type system, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.

[49]

X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Woodhead Publishing Limited, Chichester, 2008.

[50]

X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud., 7 (2000), 307-328. 

[51]

X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl., 65 (1996), 233-250.  doi: 10.1016/S0304-4149(96)00109-3.

[52]

F. Morillas and J. Valero, A Peano's theorem and attractors for lattice dynamical systems, International J. Bifur. Chaos, 19 (2009), 557-578.  doi: 10.1142/S0218127409023196.

[53]

O. MisiatsO. Stanzhytskyi and N. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, Journal of Theoretical Probability, 29 (2016), 996-1026.  doi: 10.1007/s10959-015-0606-z.

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S. E. A. Mohammed, Stochastic Functional Differential Equations, Longman, New York, 1984.

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J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.

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T. Naito, On autonomous linear retarded equations in abstract phase for infinite retardations, J. Differential Equations, 21 (1976), 297-315.  doi: 10.1016/0022-0396(76)90124-8.

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T. Naito, On linear autonomous retarded equations in abstract phase for infinite delay, J. Differential Equations, 33 (1979), 74-91.  doi: 10.1016/0022-0396(79)90081-0.

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M. ReissM. Riedle and O. van Gaans, Delay differential equations driven by Levy processes: Stationarity and Feller properties, Stoch. Process. Appl., 116 (2006), 1409-1432.  doi: 10.1016/j.spa.2006.03.002.

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X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dyn. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

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show all references

References:
[1]

V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631-637.  doi: 10.1142/S0218127494000459.

[2]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[3]

P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.

[4]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, International J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[5]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[7]

J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501.

[8]

W. J. Beyn and S. Y. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 15 (2003), 485-515.  doi: 10.1023/B:JODY.0000009745.41889.30.

[9]

Z. BrzezniakM. Ondrejat and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differential Equations, 260 (2016), 4157-4179.  doi: 10.1016/j.jde.2015.11.007.

[10]

Z. BrzezniakE. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Annals of Probability, 45 (2017), 3145-3201.  doi: 10.1214/16-AOP1133.

[11]

O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), 1-23.  doi: 10.1214/17-EJP122.

[12]

T. CaraballoM. J. Garrido-Atienza and B. Schmalfuss, Exponential stability of stationary solutions for semilinear stochastic evolution equations with delays, Discret. Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.

[13]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[14]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[15]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[16]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, I, II,, IEEE Trans. Circuits Systems, 42 (1995), 746–751. doi: 10.1109/81.473583.

[17]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 49 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[18]

S. N. ChowJ. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Computational Dynamics, 4 (1996), 109-178. 

[19]

S. N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764-1781.  doi: 10.1137/S0036139994261757.

[20]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems, 40 (1993), 147-156.  doi: 10.1109/81.222795.

[21]

L. O. Chua and Y. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.

[22] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[23]

J. Eckmann and M. Hairer, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, 14 (2001), 133-151.  doi: 10.1088/0951-7715/14/1/308.

[24]

C. E. Elmer and E. S. Van Vleck, Analysis and computation of traveling wave solutions of bistable differential-difference equations, Nonlinearity, 12 (1999), 771-798.  doi: 10.1088/0951-7715/12/4/303.

[25]

C. E. Elmer and E. S. Van Vleck, Traveling waves solutions for bistable differential-difference equations with periodic diffusion, SIAM J. Appl. Math., 61 (2001), 1648-1679.  doi: 10.1137/S0036139999357113.

[26]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.

[27]

A. Es-SarhirM. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200. 

[28]

M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stochastics and Dynamics, 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.

[29]

K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[30]

J. K. Hale, Functional differential equations with infinite delays, J. Math. Anal. Appl., 48 (1974), 276-283.  doi: 10.1016/0022-247X(74)90233-9.

[31]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[32]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.

[33]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[34]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.

[35]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.

[36]

Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[37]

K. Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.

[38]

R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.

[39]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrodinger equation, J. Differential Equations, 217 (2005), 88-123.  doi: 10.1016/j.jde.2005.06.002.

[40]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.

[41]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.

[42]

J. Kim, On the stochastic Burgers equation with polynomial nonlinearity in the real line, Discrete Continuous Dynam. Systems - B, 6 (2006), 835-866.  doi: 10.3934/dcdsb.2006.6.835.

[43]

J. Kim, On the stochastic Benjamin-Ono equation, J. Differential Equations, 228 (2006), 737-768.  doi: 10.1016/j.jde.2005.11.005.

[44]

J. Kim, Periodic and invariant measures for stochastic wave equations, Electronic Journal of Differential Equations, 2004 (2004), 1-30. 

[45]

J. Kim, Invariant measures for a stochastic nonlinear Schrodinger equation, Indiana University Mathematics Journal, 55 (2006), 687-717.  doi: 10.1512/iumj.2006.55.2701.

[46] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986. 
[47] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Boston, 1993. 
[48]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type system, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.

[49]

X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Woodhead Publishing Limited, Chichester, 2008.

[50]

X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud., 7 (2000), 307-328. 

[51]

X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl., 65 (1996), 233-250.  doi: 10.1016/S0304-4149(96)00109-3.

[52]

F. Morillas and J. Valero, A Peano's theorem and attractors for lattice dynamical systems, International J. Bifur. Chaos, 19 (2009), 557-578.  doi: 10.1142/S0218127409023196.

[53]

O. MisiatsO. Stanzhytskyi and N. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, Journal of Theoretical Probability, 29 (2016), 996-1026.  doi: 10.1007/s10959-015-0606-z.

[54]

S. E. A. Mohammed, Stochastic Functional Differential Equations, Longman, New York, 1984.

[55]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.

[56]

T. Naito, On autonomous linear retarded equations in abstract phase for infinite retardations, J. Differential Equations, 21 (1976), 297-315.  doi: 10.1016/0022-0396(76)90124-8.

[57]

T. Naito, On linear autonomous retarded equations in abstract phase for infinite delay, J. Differential Equations, 33 (1979), 74-91.  doi: 10.1016/0022-0396(79)90081-0.

[58]

M. ReissM. Riedle and O. van Gaans, Delay differential equations driven by Levy processes: Stationarity and Feller properties, Stoch. Process. Appl., 116 (2006), 1409-1432.  doi: 10.1016/j.spa.2006.03.002.

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