Abstract
We study the effect of Gaussian perturbations on a hyperbolic partial differential equation with double characteristics in two spatial dimensions. The coefficients of our partial differential operator depend polynomially on the space variables, while the noise is additive, white in time and coloured in space. We provide a sufficient condition on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Our approach is based on explicit computations for the fundamental solution of the partial differential operator and its Fourier transform.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdeljawad, A., Ascanelli, A., Coriasco, S.: Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on \({\mathbb{R} }^n\). Monatsh. Math. 192, 1–38 (2020)
Ascanelli, A., Coriasco, S.S., Süß, A.: Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded coefficients. Nonlinear Anal. Theory Methods Appl. 189, 111–574 (2019)
Ascanelli, A., Coriasco, S., Süß, A.: Random-field solutions of weakly hyperbolic stochastic partial differential equations with polynomially bounded coefficients. J. Pseudo-Differ. Oper. Appl. 11(1), 387–424 (2020)
Ascanelli, A., Süß, A.: Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients. Stochast. Processes Appl. 128, 2605–2641 (2018)
Bernardi, E., Nishitani, T.: On the Cauchy problem for noneffectively hyperbolic operators: the Gevrey 4 well-posedness. Kyoto J. Math. 51, 767–810 (2011)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, II Cambridge University Press, Cambridge (2014)
Dalang, R. C.: Extending martingale measure stochastic integral with application to spatially homogenous S.P.D.E.’s. Electr. J. Probab. 4, 1–29 (1999)
Dalang, R.C., Frangos, N.E.: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26(1), 187–212 (1998)
Dalang, R. C.: Some non-linear S.P.D.E.’s that are second order in time. Electron. J. Probab. 8, 1–21 (2003)
Dalang, R.C., Quer-Sardanyons, L.: Stochastic integral for spde’s: a comparison. Expo. Math. 29, 67–109 (2011)
Hörmander, L.: The Cauchy problem for differential equations with double characteristics. J. Anal. Math. 32, 118–196 (1977)
Hörmander, L.: Quadratic hyperbolic operators microlocal analysis and applications (L. Cattabriga and L. Rodino eds). Lecture Notes Math. 1495, 118–160 (1991)
Ivrii, V.J., Petkov, V.M.: Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations. Uspehi. Mat. Nauk 29, 3–70 (1974)
Orsingher, E.: Randomly forced vibrations of a string. Annales de l’I.H.P. Probabilités et Statistiques 18, 367–394 (1982)
Peszat, S.: The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2, 383–394 (2002)
Sanz-Solé, M., Süß, A.: The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity. Electron. J. Probab. 18, 1–28 (2013)
Walsh, J.B., An introduction to stochastic partial differential equations, Ecole d’Ete de Prob. de St-Flour XIV 1984, Lect. Notes in Math 1180, Springer Verlag (1986)
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alexey Karapetyants.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bernardi, E., Lanconelli, A. On a Class of Stochastic Hyperbolic Equations with Double Characteristics. J Fourier Anal Appl 29, 2 (2023). https://doi.org/10.1007/s00041-022-09987-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-022-09987-7