Advertisement

Stationarity of the Solution for the Semilinear Stochastic Integral Equation on the Whole Real Line

  • Bijan Z. ZangenehEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

In this article we prove the stationarity of the solution of the H-valued integral equation
$$X(t) =\displaystyle\int _{ -\infty }^{t}U(t - s)f(X(s))\mathrm{d}s + V (t),$$
where H is a real separable Hilbert space. In this equation, U(t) is a semigroup generated by a strictly negative definite, self-adjoint unbounded operator A, such that A  − 1 is compact and f is of monotone type and is bounded by a polynomial and V (t) is a cadlag adapted stationary process.

Keywords

Integral Equation Mild Solution Separable Hilbert Space Energy Inequality Galerkin Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Albeverio, S., Mandrekar, V., Rudiger, B.: Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise. Stoch. Process. Appl. 119, 835–863 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Albeverio, S., Rudiger, B.: Stochastic Integrals and the Lévy-Itô decomposition theorem on separable Banach spaces. Stoch. Anal. Appl. 23(2), 217–253 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via dirichlet forms. Probab. Theor. Rel. Fields 89, 347–386 (1991)zbMATHCrossRefGoogle Scholar
  4. 4.
    Applebaum, D.: Levy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  5. 5.
    Alós, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75(3), 129–152 (2003)MathSciNetGoogle Scholar
  6. 6.
    Caraballo, T., Garrido-Atienza, M.J., Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74(11), 3671–3684 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Da Prato, G., Zabczyk, J.: Ergodicity for infinite-dimensional systems. In: London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996)Google Scholar
  8. 8.
    Da Prato, G., Zabcyk, J.: Stochastic equations in Infinite dimensions. In: Encyclopedia of Mathematics and its Applications, vol. 45. Cambridge University Press, Cambridge (1992)Google Scholar
  9. 9.
    Duncan, T.E., Hu, Y., Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion I Theory. SIAM J. Contr. Optim. 38(2), 582–612 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ferrante, M., Rovira, C.: Convergence of delay differential equations driven by fractional Brownian motion. J. Evol. Equat. 10(4), 761–783 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Funaki, T.: Random motion of strings and related stochastic evolution equations. Nagoya Math. 89, 129–193 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hamedani, H.D., Zangeneh, B.Z.: The existence, uniqueness, and measurability of a stopped semilinear integral equation. Stoch. Anal. Appl. 25(3), 493–518 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Iscoe, I., Marcus, M.B., McDonald, D., Talagrand, M., Zinn, J.: Continuity of l 2-valued Ornstein-Uhlenbeck Process. Ann. Probab. 18(1), 68–84 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Iwata, K.: An infinite dimensional stochastic differential equation with state space C(R). Probab. Theor. Rel. Fields 5743, 141–159 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jona-Lasinio, P., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101, 409–436 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kolmogorov, A.: Zur umker barkeit der statistischen naturgesetze. Math. Ann. 113, 766–772 (1937)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Marcus, R.: Parabolic Ito equations. Trans. Am. Math. Soc. 198, 177–190 (1974)zbMATHGoogle Scholar
  18. 18.
    Marcus, R.: Parabolic Ito equations with monotone non-linearities. Funct. Anal. 29, 275–286 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Marcus, R.: Stochastic diffusion on an unbounded domain. Pacific J. Math. 84(1), 143–153 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202(1), 277–305 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Miyahara, Y.: Infinite dimensional Langevin equation and Fokker-Planck equation. Nagoya Math. J. 81, 177–223 (1981)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Nualart, D., Rascanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53(1), 55–81 (2002)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Levy noise. An evolution equation approach. In: Encyclopedia of Mathematics and its Applications, vol. 113. Cambridge University Press, Cambridge, (2007)Google Scholar
  24. 24.
    Riedle, M., van Gaans, O.: Stochastic integration for Lévy processes with values in Banach spaces. Stoch. Process. Appl. 119 1952–1974 (2009)zbMATHCrossRefGoogle Scholar
  25. 25.
    Tindel, S., Tudor, C.A., Viens, F.: Stochastic evolution equations with fractional Brownian motion. Probab. Theor. Rel. Fields 127(2), 186–204 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Walsh, J.B.: An introduction to stochastic partial differential equations. Lect. Notes Math. 1180, 266–439 (1986)Google Scholar
  27. 27.
    Zamani, S.: Reaction-diffusion equations with polynomial drifts driven by fractional Brownian motions. Stoch. Anal. Appl. 28(6), 1020–1039 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Zangeneh, B.Z.: An energy-type inequality. Math. Inequal. Appl. 1(3), 453–461 (1998)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zangeneh, B.Z.: Galerkin approximations for a semilinear stochastic integral equation. Sci. Iran. 4(1–2), 8–11 (1997)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zangeneh, B.Z.: Semilinear stochastic evolution equations with monotone nonlinearities. Stoch. Stoch. Rep. 53, 129–174 (1995)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Zangeneh, B.Z.: Existence and uniqueness of the solution of a semilinear stochastic evolution equation on the whole real line. In: Seminar on Stochastic Processes. Birkhãuser, Boston (1992)Google Scholar
  32. 32.
    Zangeneh, B.Z.: Measurability of the solution of a semilinear evolution equation. In: Seminar on Stochastic Processes. Birkhãuser, Boston (1990)Google Scholar
  33. 33.
    Zangeneh, B.Z.: Semilinear Stochastic Evolution Equations. Ph.D. thesis, University of British Columbia, Vancouver, B.C. Canada (1990)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIran

Personalised recommendations