# American Institute of Mathematical Sciences

doi: 10.3934/eect.2022005
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## Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients

 1 Department of Physics and Mathematics, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine 2 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284, USA 3 Department of Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

* Corresponding author: Oleksandr Misiats

Received  March 2021 Revised  November 2021 Early access February 2022

In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.

Citation: Andriy Stanzhytsky, Oleksandr Misiats, Oleksandr Stanzhytskyi. Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients. Evolution Equations and Control Theory, doi: 10.3934/eect.2022005
##### References:
 [1] A. Anguraj and A. Vinodkumar, Existence, uniqueness and stability of impulsive stochactic partial neutral functional differential equations with infinite delays, J. Appl. Math. Informatics, 28 (2010), 739-751. [2] B. Boufoussi, S. Hajji and E. Lakhel, Time-dependent neutral stochastic functional differential equations driven by a fractional Brownian motion, Commun. Stoch. Anal., 10 (2016), Article 1, 1–12. doi: 10.31390/cosa. 10.1.01. [3] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101. [4] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829. [5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223. [6] D. Dawson, Stochastic evolution equations, Math. Biosci., 15 (1972), 287-316.  doi: 10.1016/0025-5564(72)90039-9. [7] L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019. [8] T. E. Govindan, Mild solutions of neutral stochastic partial functional differential equations, Int. J. Stoch. Anal., (2011), Art. ID 186206, 13 pp. doi: 10.1155/2011/186206. [9] P. Grisvald, Commutativite de deux foncteurs d'interpolation et applications, J. Math. Pures. Appl., 45 (1966), 207-290. [10] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. [12] M. Hieber, O. Misiats and O. Stanzhytskyi, On the bidomain equations driven by stochastic forces, Discrete Contin. Dyn. Syst., 40 (2020), 6159-6177.  doi: 10.3934/dcds.2020274. [13] F. Jiang and Y. Shen, A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 61 (2011), 1590-1594.  doi: 10.1016/j.camwa.2011.01.027. [14] K. Kenzhebaev, A. Stanzhytskyi and A. Tsukanova., Existence and uniqueness results, the markovian property of solutions for a neutral delay stochastic reaction-diffusion equation in entire space, Dynamic Systems and Applications, 28 (2019), 19-46.  doi: 10.12732/dsa.v28i1.2. [15] N. Kryloff and N. Bogoliouboff, La théorie générale de la mesure dans son application à l'étude des systèemes dynamiques de la mécanique non linéaire, Ann. of Math. (2), 38 (1937), 65-113.  doi: 10.2307/1968511. [16] Q. Li and M. Wei., Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay, Evol. Equ. Control Theory, 9 (2020), 753-772.  doi: 10.3934/eect.2020032. [17] K. Liu, Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3915-3934.  doi: 10.3934/dcdsb.2018117. [18] J. Luo, Exponential stability for stochastic neutral partial functional differential equations, J. Math. Anal. Appl., 355 (2009), 414-425.  doi: 10.1016/j.jmaa.2009.02.001. [19] R. Manthey and T. Zausinger, Stochastic evolution equations in $L^{2\nu}_\rho$, Stochastics Stochastics Rep., 66 (1999), 37-85.  doi: 10.1080/17442509908834186. [20] J. Maxwell, On the dynamical theory of gases, Philosophical transactions of the Royal Society of London, 157 (1867), 49-88. [21] O. Misiats, V. Mogilova and O. Stanzhytskyi, Invariant measure for stochastic functional differential equations in Hilbert spaces, arXiv: 2011.07034, 2020. [22] O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theoret. Probab., 29 (2016), 996-1026.  doi: 10.1007/s10959-015-0606-z. [23] O. Misiats, O. Stanzhytskyi and N. K. Yip, Asymptotic analysis and homogenization of invariant measures, Stoch. Dyn., 19 (2019), 1950015, 27 pp. doi: 10.1142/S0219493719500151. [24] O. Misiats, O. Stanzhytskyi and N. K. Yip, Invariant measures for stochastic reaction-diffusion equations with weakly dissipative nonlinearities, Stochastics, 92 (2020), 1197-1222.  doi: 10.1080/17442508.2019.1691212. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] D. Ruan and J. Luo., The existence, uniqueness, and controllability of neutral stochastic delay partial differential equations driven by standard Brownian motion and fractional Brownian motion, Discrete Dyn. Nat. Soc., (2018), Art. ID 7502514, 11 pp. doi: 10.1155/2018/7502514. [27] A. M. Samoilenko, N. I. Mahmudov and A. N. Stanzhytskyi, Existence, uniqueness, and controllability results for neutral FSDES in Hilbert spaces, Dynam. Systems Appl., 17 (2008), 53-70. [28] O. M. Stanzhytskyi, Investigation of invariant sets of Itô stochastic systems by means of Lyapunov functions, Ukraïn. Mat. Zh., 53 (2001), 282–285. doi: 10.1023/A: 1010437625118. [29] O. M. Stanzhytskyi, Investigation of the exponential dichotomy of Itô stochastic systems by means of quadratic forms, Ukraïn. Mat. Zh., 53 (2001), 1545–1555. doi: 10.1023/A: 1015259031308. [30] O. Stanzhytskyi, V. V. Mogilova and A. O. Tsukanova, On comparison results for neutral stochastic differential equations of reaction-diffusion type in $L_2(\Bbb mathbb{R}^{d})$, In Modern Mathematics and Mechanics, Underst. Complex Syst., 351–395, Springer, 2019. [31] E. F. Tsar'kov, Random Perturbations of Functional Differential Equations, Zinatne, Riga, 1989.

show all references

##### References:
 [1] A. Anguraj and A. Vinodkumar, Existence, uniqueness and stability of impulsive stochactic partial neutral functional differential equations with infinite delays, J. Appl. Math. Informatics, 28 (2010), 739-751. [2] B. Boufoussi, S. Hajji and E. Lakhel, Time-dependent neutral stochastic functional differential equations driven by a fractional Brownian motion, Commun. Stoch. Anal., 10 (2016), Article 1, 1–12. doi: 10.31390/cosa. 10.1.01. [3] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101. [4] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829. [5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223. [6] D. Dawson, Stochastic evolution equations, Math. Biosci., 15 (1972), 287-316.  doi: 10.1016/0025-5564(72)90039-9. [7] L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019. [8] T. E. Govindan, Mild solutions of neutral stochastic partial functional differential equations, Int. J. Stoch. Anal., (2011), Art. ID 186206, 13 pp. doi: 10.1155/2011/186206. [9] P. Grisvald, Commutativite de deux foncteurs d'interpolation et applications, J. Math. Pures. Appl., 45 (1966), 207-290. [10] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. [12] M. Hieber, O. Misiats and O. Stanzhytskyi, On the bidomain equations driven by stochastic forces, Discrete Contin. Dyn. Syst., 40 (2020), 6159-6177.  doi: 10.3934/dcds.2020274. [13] F. Jiang and Y. Shen, A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 61 (2011), 1590-1594.  doi: 10.1016/j.camwa.2011.01.027. [14] K. Kenzhebaev, A. Stanzhytskyi and A. Tsukanova., Existence and uniqueness results, the markovian property of solutions for a neutral delay stochastic reaction-diffusion equation in entire space, Dynamic Systems and Applications, 28 (2019), 19-46.  doi: 10.12732/dsa.v28i1.2. [15] N. Kryloff and N. Bogoliouboff, La théorie générale de la mesure dans son application à l'étude des systèemes dynamiques de la mécanique non linéaire, Ann. of Math. (2), 38 (1937), 65-113.  doi: 10.2307/1968511. [16] Q. Li and M. Wei., Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay, Evol. Equ. Control Theory, 9 (2020), 753-772.  doi: 10.3934/eect.2020032. [17] K. Liu, Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3915-3934.  doi: 10.3934/dcdsb.2018117. [18] J. Luo, Exponential stability for stochastic neutral partial functional differential equations, J. Math. Anal. Appl., 355 (2009), 414-425.  doi: 10.1016/j.jmaa.2009.02.001. [19] R. Manthey and T. Zausinger, Stochastic evolution equations in $L^{2\nu}_\rho$, Stochastics Stochastics Rep., 66 (1999), 37-85.  doi: 10.1080/17442509908834186. [20] J. Maxwell, On the dynamical theory of gases, Philosophical transactions of the Royal Society of London, 157 (1867), 49-88. [21] O. Misiats, V. Mogilova and O. Stanzhytskyi, Invariant measure for stochastic functional differential equations in Hilbert spaces, arXiv: 2011.07034, 2020. [22] O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theoret. Probab., 29 (2016), 996-1026.  doi: 10.1007/s10959-015-0606-z. [23] O. Misiats, O. Stanzhytskyi and N. K. Yip, Asymptotic analysis and homogenization of invariant measures, Stoch. Dyn., 19 (2019), 1950015, 27 pp. doi: 10.1142/S0219493719500151. [24] O. Misiats, O. Stanzhytskyi and N. K. Yip, Invariant measures for stochastic reaction-diffusion equations with weakly dissipative nonlinearities, Stochastics, 92 (2020), 1197-1222.  doi: 10.1080/17442508.2019.1691212. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] D. Ruan and J. Luo., The existence, uniqueness, and controllability of neutral stochastic delay partial differential equations driven by standard Brownian motion and fractional Brownian motion, Discrete Dyn. Nat. Soc., (2018), Art. ID 7502514, 11 pp. doi: 10.1155/2018/7502514. [27] A. M. Samoilenko, N. I. Mahmudov and A. N. Stanzhytskyi, Existence, uniqueness, and controllability results for neutral FSDES in Hilbert spaces, Dynam. Systems Appl., 17 (2008), 53-70. [28] O. M. Stanzhytskyi, Investigation of invariant sets of Itô stochastic systems by means of Lyapunov functions, Ukraïn. Mat. Zh., 53 (2001), 282–285. doi: 10.1023/A: 1010437625118. [29] O. M. Stanzhytskyi, Investigation of the exponential dichotomy of Itô stochastic systems by means of quadratic forms, Ukraïn. Mat. Zh., 53 (2001), 1545–1555. doi: 10.1023/A: 1015259031308. [30] O. Stanzhytskyi, V. V. Mogilova and A. O. Tsukanova, On comparison results for neutral stochastic differential equations of reaction-diffusion type in $L_2(\Bbb mathbb{R}^{d})$, In Modern Mathematics and Mechanics, Underst. Complex Syst., 351–395, Springer, 2019. [31] E. F. Tsar'kov, Random Perturbations of Functional Differential Equations, Zinatne, Riga, 1989.
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