Advertisement

Random-field solutions of weakly hyperbolic stochastic partial differential equations with polynomially bounded coefficients

  • Alessia AscanelliEmail author
  • Sandro Coriasco
  • André Süss
Article

Abstract

We study random-field solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider linear equations under suitable hyperbolicity hypotheses, and we provide conditions on the initial data and on the stochastic term, namely, on the associated spectral measure, so that these kind of solutions exist in suitably chosen functional classes. We also give a regularity result for the expected value of the solution.

Keywords

Hyperbolic stochastic partial differential equations Random-field solutions Variable coefficients Fundamental solution Fourier integral operators 

Mathematics Subject Classification

Primary 35L10 60H15 Secondary 35L40 35S30 

Notes

Acknowledgements

The authors have been supported by the INdAM-GNAMPA grant 2014 “Equazioni Differenziali a Derivate Parziali di Evoluzione e Stocastiche” (Coordinator: S. Coriasco, Dep. of Mathematics “G. Peano”, University of Turin) and by the INdAM-GNAMPA grant 2015 “Equazioni Differenziali a Derivate Parziali di Evoluzione e Stocastiche” (Coordinator: A. Ascanelli, Dep. of Mathematics and Computer Science, University of Ferrara). The third author has been partially supported by the grant MTM 2015-65092-P by the Secretaria de estado de investigación, desarrollo e innovación, Ministerio de Economía y Competitividad. Thanks are due, for very useful discussions and observations, to R. Denk, T. Hartung, M. Oberguggenberger, S. Pilipović, E. Priola, D. Seleši, and I. Witt.

References

  1. 1.
    Abdeljawad, A., Ascanelli, A., Coriasco, S.: Deterministic and Stochastic Cauchy problems for a class of weakly hyperbolic operators on \(R^n\). Preprint, arXiv:1810.05009 (2018)
  2. 2.
    Ascanelli, A., Cappiello, M.: Log-Lipschitz regularity for \(SG\) hyperbolic systems. J. Differ. Equ. 230, 556–578 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ascanelli, A., Cappiello, M.: The Cauchy problem for finitely degenerate hyperbolic equations with polynomial coefficients. Osaka J. Math. 47(2), 423–438 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ascanelli, A., Coriasco, S.: Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on \(\mathbb{R}^n\). J. Pseudo-Differ. Oper. Appl. 6, 521–565 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ascanelli, A., Coriasco, S., Süß, A.: On temperate distributions decaying at infinity. In: Oberguggenberger, M., Toft, J., Vindas, J., Wahlberg, P. (eds.) Generalized Functions and Fourier Analysis (Operator Theory: Advances and Applications), vol. 260, pp. 1–18. Springer, Berlin (2017)Google Scholar
  6. 6.
    Ascanelli, A., Coriasco, S., Süß, A.: Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded coefficients. Preprint, arXiv:1610.01208 (2018)
  7. 7.
    Ascanelli, A., Süß, A.: Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients. Stochast. Processes Appl. 128, 2605–2641 (2018)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)zbMATHGoogle Scholar
  9. 9.
    Cicognani, M., Zanghirati, L.: Analytic regularity for solutions of nonlinear weakly hyperbolic equations. Boll. Un. Mat. Ital 11–B(7), 643–679 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cicognani, M., Zanghirati, L.: The Cauchy problem for nonlinear hyperbolic equations with Levi’s condition. Bull. Sci. Math. 123, 413–435 (1999)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Conus, D., Dalang, R.C.: The non-linear stochastic wave equation in high dimensions. Electron. J. Probabil. 13, 629–670 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cordero, E., Nicola, F., Rodino, L.: On the global boundedness of Fourier integral operators. Ann. Glob. Anal. Geom. 38, 373–398 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cordes, H.O.: The Technique of Pseudodifferential Operators. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  14. 14.
    Coriasco, S.: Fourier integral operators in \(SG\) classes I. Composition theorems and action on \(SG\) Sobolev spaces. Rend. Sem. Mat. Univ. Pol. Torino 57(4), 249–302 (1999)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Coriasco, S.: Fourier integral operators in \(SG\) classes II. Application to \(SG\) hyperbolic Cauchy problems. Ann. Univ. Ferrara 47, 81–122 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Coriasco, S., Maniccia, L.: On the spectral asymptotics of operators on manifolds with ends. Abstr. Appl. Anal. 2013, Article ID 909782 (2013)Google Scholar
  17. 17.
    Coriasco, S., Rodino, L.: Cauchy problem for \(SG\)-hyperbolic equations with constant multiplicities. Ric. di Matematica 48(Suppl.), 25–43 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dalang, R.C., Mueller, C.: Some non-linear S.P.D.E.’s that are second order in time. Electron. J. Probab. 8(1), 1–21 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogeneous SPDEs. Electron. J. Probab. 4, 1–29 (1999)MathSciNetGoogle Scholar
  20. 20.
    Dalang, R.C., Frangos, N.: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26(1), 187–212 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dalang, R.C., Quer-Sardanyons, L.: Stochastic integral for spde’s: a comparison. Expositiones Mathematicae 29, 67–109 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    DaPrato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 45. Cambridge University Press, Cambridge (2008)Google Scholar
  23. 23.
    Grigis, A., Sjöstrand, J.: Microlocal Analysis for Differential Operators. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  24. 24.
    Kumano-go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1981)zbMATHGoogle Scholar
  25. 25.
    Melrose, R.: Geometric Scattering Theory (Stanford Lectures). Cambridge University Press, Cambridge (1995)Google Scholar
  26. 26.
    Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, Cambridge (1973)zbMATHGoogle Scholar
  27. 27.
    Mizohata, S.: On the Cauchy Problem. Academic Press Inc, Cambridge (1985)zbMATHGoogle Scholar
  28. 28.
    Morimoto, Y.: Fundamental solutions for a hyperbolic equation with involutive characteristics of variable multiplicity. Commun. Partial Differ. Equ. 4(6), 609–643 (1979)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Oberguggenberger, M., Schwarz, M.: Fourier integral operators in stochastic structural analysis. In: Proceedings of the 12th International Probabilistic Workshop (2014)Google Scholar
  30. 30.
    Parenti, C.: Operatori pseudodifferenziali in \({\mathbb{R}}^n\) e applicazioni. Ann. Mat. Pura Appl. 93, 359–389 (1972)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Peszat, S.: The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evolut. Equ. 2(3), 383–394 (2002)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ruzhansky, M., Sugimoto, M.: Global \(L^2\) boundedness theorems for a class of Fourier integral operators. Commun. Partial Differ. Equ. 31, 547–569 (2006)zbMATHGoogle Scholar
  33. 33.
    Sanz-Solé, M., Süß, A.: The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity. Electron. J. Probab. 18, Paper no 64 (2013)Google Scholar
  34. 34.
    Schwartz, L.: Théorie des Distributions, 2nd edn. Hermann, Paris (2010)Google Scholar
  35. 35.
    Taniguchi, K.: Multi-products of Fourier integral operators and the fundamental solution for a hyperbolic system with involutive characteristics. Osaka J. Math. 21, 169–224 (1984)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tindel, S.: Spdes with pseudo-differential generators: the existence of a density. Applicationes Matematicae 27, 287–308 (2000)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Walsh, J.B.: École d’été de Probabilités de Saint Flour XIV, 1984, volume 1180 of Lecture Notes in Math, chapter An Introduction to Stochastic Partial Differential Equations. Springer, Berlin (1986)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica ed InformaticaUniversità di FerraraFerraraItaly
  2. 2.Dipartimento di Matematica “G. Peano”Università degli Studi di TorinoTorinoItaly

Personalised recommendations