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Trotter-Kato approximations of stochastic neutral partial functional differential equations

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Abstract

In this paper, a stochastic neutral partial functional differential equation is studied in real separable Hilbert spaces. The aim here is to introduce Trotter-Kato approximations of mild solutions for this class of equations. As an application, a classical limit theorem on the dependence of such equations on a parameter is obtained. Moreover, weak convergence of probability measures induced by the Trotter-Kato approximate mild solutions is established. An example is included at the end.

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References

  1. N. U. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stochastic Proc. Appl., 60 (1995), 65-85.

    Article  MathSciNet  Google Scholar 

  2. W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, Vol. 96, Second Edition, Birkhäuser-Verlag, Basel (2011).

  3. Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Reports, 61 (1997), 245-295.

    Article  MathSciNet  Google Scholar 

  4. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge (1992).

    Book  Google Scholar 

  5. I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, Berlin (1972).

    Book  Google Scholar 

  6. T. E. Govindan, Autonomous semilinear stochastic Volterra integrodifferential equations in Hilbert spaces, Dynamic Systems Appl., 3 (1994), 51-74.

    MathSciNet  MATH  Google Scholar 

  7. T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77 (2005), 139-154.

    Article  MathSciNet  Google Scholar 

  8. T. E. Govindan, Trotter-Kato approximations of semilinear stochastic evolution equations, Boletin Soc. Mat. Mexicana, 12 (2006), 109-120.

    MathSciNet  MATH  Google Scholar 

  9. T. E. Govindan, On Trotter-Kato approximations of semilinear stochastic evolution equations in infinite dimensions, Statis. Probab. Letters, 96 (2015), 299-306.

    Article  MathSciNet  Google Scholar 

  10. T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling Series, Vol. 79, Springer (2016).

    Book  Google Scholar 

  11. A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 90 (1982), 12-44.

    Article  MathSciNet  Google Scholar 

  12. D. Kannan and A. T. Bharucha-Reid, On a stochastic integrodifferential evolution equation of Volterra type, J. Integral Eqns., 10 (1985), 351-379.

    MATH  Google Scholar 

  13. M. C. Kunze and J. M. A. M. van Neerven, Approximating the coefficients in semilinear stochastic partial differential equations, J. Evolution Eqns., 11 (2011), 577-604.

    Article  MathSciNet  Google Scholar 

  14. M. C. Kunze and J. M. A. M. van Neerven, Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations, J. Differential Eqns., 253 (2012), 1036-1068.

    Article  MathSciNet  Google Scholar 

  15. K. Liu, Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics Stochastics Reports, 63 (1998), 1-26.

    Article  MathSciNet  Google Scholar 

  16. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin (1983).

    Book  Google Scholar 

  17. T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stochas. Anal. Appl., 16 (1998), 965-975.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author sincerely thanks an anonymous referee for reading the paper very carefully and for pointing out some minor corrections that led to the improvement of the paper. The author also thanks SIP-IPN for partial financial support.

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Authors and Affiliations

  1. Department of Mathematics, ESFM, National Polytechnic Institute, 07738, Mexico City, Mexico

    T. E. Govindan

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  1. T. E. Govindan
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Correspondence to T. E. Govindan.

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Communicated by Rahul Roy.

Dedicated to my mother Mrs. G. Suseela.

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Govindan, T.E. Trotter-Kato approximations of stochastic neutral partial functional differential equations. Indian J Pure Appl Math 52, 822–836 (2021). https://doi.org/10.1007/s13226-021-00146-0

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  • DOI: https://doi.org/10.1007/s13226-021-00146-0

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