Advertisement

Pathwise uniqueness for stochastic evolution equations with Hölder drift and stable Lévy noise

  • Desheng Yang
Article
  • 46 Downloads

Abstract

We prove the pathwise uniqueness of solutions to stochastic evolution equations in Hilbert spaces with the \(\alpha \)-stable Lévy noise and a bounded \(\beta \)-Hölder continuous drift term. The proof is based on the regularity results of resolvent equations associated to Kolmogorov operators.

Keywords

Pathwise uniqueness Stocastic PDEs Stable process Kolmogorov operators 

Mathematics Subject Classification

60H15 60J75 60J35 35R15 

References

  1. 1.
    Cerrai, S., Da Prato, G.: Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations. J. Evol. Equ. 12, 83–98 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cerrai, S., Da Prato, G., Flandoli, F.: Pathwise uniqueness for stochastic reaction–diffusion equations in Bnanch spaces with an Hölder drift component. Stoch. Part. Differ. Equ. Anal. Comput. 1, 507–551 (2013)zbMATHGoogle Scholar
  3. 3.
    Da Prato, G., Flandoli, F.: Pathwise uniqueness for a class of SDEs in Hilbert spaces and applications. J. Funct. Anal. 259, 243–267 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41, 3306–3344 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theor. Probab. 28, 1571–1600 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Models. Saint Flour Summer School Lectures 2010. Lecture Notes in Mathematics 2015. Springer, Berlin (2011)Google Scholar
  7. 7.
    Marinelli, C., Prévôt, C., Röckner, M.: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal. 2582(2), 616–649 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Priola, E.: On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136(3), 271–295 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Priola, E., Zabczyk, J.: Structural properties of semilinear SPDEs driven by cylindrical stable processes. Probab. Theory Relat. Fields 149, 97–137 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Priola, E.: Pathwise uniqueness for singular SDEs driven by stable processes. Osaka J. Math. 49, 421–447 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Priola, E., Tracá, S.: On the Cauchy problem for non-local Ornstein–Uhlenbeck operators. Noninear Anal. 131, 182–205 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tanaka, H., Tsuchiya, M., Watanabe, S.: Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14, 73–92 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Veretennikov, A.J.: Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 153(3), 434–452 (1980)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina

Personalised recommendations