SPDEs with Volterra Noise

  • Petr Čoupek
  • Bohdan MaslowskiEmail author
  • Jana Šnupárková
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)


Recent results on linear stochastic partial differential equations driven by Volterra processes with linear or bilinear noise are briefly reviewed and partially extended. In the linear case, existence and regularity properties of stochastic convolution integral are established and the results are applied to 1D linear parabolic PDEs with boundary noise of Volterra type. For the equations with bilinear noise, existence and large time behaviour of solutions are studied.


Volterra process Rosenblatt process Stochastic evolution equation Additive noise Bilinear noise 

2010 Mathematics Subject Classification

60H15 60G22 



The authors are grateful to the anonymous referee for their valuable suggestions. The first author was supported by the Charles University grant GAUK No. 322715 and SVV 2016 No. 260334. The second author was supported by the Czech Science Foundation grant GAČR No. 15-08819S.


  1. 1.
    Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29(2), 766–801 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alòs, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75(3), 129–152 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Čoupek, P., Maslowski, B.: Stochastic evolution equations with Volterra noise. Stoch. Proc. Appl. 127(3), 877–900 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Oxford University Press, Encyclopedia of Mathematics and its Applications (2014)CrossRefGoogle Scholar
  5. 5.
    Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10(2), 177–214 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch. Dyn. 2(2), 225–250 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Stochastic equations in Hilbert spaces with a multiplicative fractional Gaussian noise. Stoch. Proc. Appl. 115(8), 1357–1383 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Stochastic linear-quadratic control for bilinear evolution equations driven by Gauss-Volterra processes (2016).Google Scholar
  9. 9.
    Lebovits, J.: Stochastic calculus with respect to Gaussian processes: Part I (2017). URL
  10. 10.
    Maslowski, B.: Stability of semilinear equations with boundary and pointwise noise. Ann. Scuola Norm. - Sci. 22(1), 55–93 (1995)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Probability and its applications (2006)zbMATHGoogle Scholar
  12. 12.
    Šnupárková, J., Maslowski, B.: Stochastic affine evolution equations with multiplicative fractional noise (2016). URL
  13. 13.
    Taqqu, M.S.: The Rosenblatt process. In: Davis, R.A., Lii, K.S., Politis, D.N. (eds.) Selected Works of Murray Rosenblatt, pp. 29–45. Springer, New York (2011)Google Scholar
  14. 14.
    Tudor, C.A.: Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12, 230–257 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Petr Čoupek
    • 1
  • Bohdan Maslowski
    • 1
    Email author
  • Jana Šnupárková
    • 2
  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  2. 2.Faculty of Chemical EngineeringUniversity of Chemistry and Technology PraguePraha 6Czech Republic

Personalised recommendations