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

  • Alessia AscanelliEmail author
  • Sandro Coriasco
  • André Süss


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.


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

Mathematics Subject Classification

Primary 35L10 60H15 Secondary 35L40 35S30 



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.


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Authors and Affiliations

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

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