Wave equation with a stable noise
Authors:
L. I. Pryhara and G. M. Shevchenko
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 96 (2018), 145-157
MSC (2010):
Primary 60H15, 35L05, 35R60, 60G52
DOI:
https://doi.org/10.1090/tpms/1040
Published electronically:
October 5, 2018
MathSciNet review:
3666878
Full-text PDF
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Additional Information
Abstract: The three-dimensional wave equation is studied in the paper. The right hand side of the equation has a symmetric $\alpha$-stable distribution. Two cases are considered, namely the cases where the perturbation is a (1) “white noise” and (2) “colored noise”. It is proved for both cases that a candidate for a solution (a function represented by the Kirchhoff formula) is a generalized solution.
References
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References
- R. M. Balan and C. A. Tudor, The stochastic wave equation with fractional noise: a random field approach, Stoch. Process. Appl. 120 (2010), no. 12, 2468–2494. MR 2728174
- I. M. Bodnarchuk, Wave equation with a stochastic measure, Teor. Imovir. Mat. Stat. 94 (2016), 1–15; English transl. in Theor. Probability and Math. Statist. 94 (2017), 1–16. MR 3553450
- R. C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212. MR 1617046
- R. C. Dalang and M. Sanz-Sole, Hölder–Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three, Mem. Amer. Math. Soc., vol. 199 (931), AMS, Providence, 2009. MR 2512755
- M. Dozzi and G. Shevchenko, Real harmonizable multifractional stable process and its local properties, Stoch. Process. Appl. 121 (2011), no. 7, 1509–1523. MR 2802463
- D. M. Gorodnya, On the existence and uniqueness of solutions of the Cauchy problem for wave equations with general stochastic measures, Theory Probab. Math. Statist. 85 (2011), 50–55; English transl. in Theor. Probability and Math. Statist. 85 (2012), 53–59. MR 2933702
- N. Kôno and M. Maejima, Hölder continuity of sample paths of some self-similar stable processes, Tokyo J. Math. 14 (1991), no. 1, 93–100. MR 1108158
- A. Millet and P.-L. Morien, On a stochastic wave equation in two space dimensions: regularity of the solution and its density, Stochastic Process. Appl. 86 (2000), no. 1, 141–162. MR 1741200
- L. Pryhara and G. Shevchenko, Stochastic wave equation in a plane driven by spatial stable noise, Mod. Stoch. Theory Appl. 3 (2016), no. 3, 237–248. MR 3576308
- L. Quer-Sardanyons and S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stoch. Process. Appl. 117 (2007), no. 10, 1448–1472. MR 2353035
- G. Samorodnitsky, Integrability of stable processes, Probab. Math. Statist. 13 (1992), no. 2, 191–204. MR 1239145
- G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994. MR 1280932
- G. M. Shevchenko, Local properties of a multifractional stable field, Theory Probab. Math. Statist. 85 (2012), 159–168. MR 2933711
- J. B. Walsh, An introduction to stochastic partial differential equations, École d’Été de Probabilités de Saint Flour XIV – 1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, 265–439. MR 876085
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Additional Information
L. I. Pryhara
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email:
pruhara7@gmail.com
G. M. Shevchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Stochastic partial differential equation,
wave equation,
LePage decomposition,
symmetric $\alpha$-stable random measure,
generalized solution,
real anisotropic fractional stable field
Received by editor(s):
January 25, 2017
Published electronically:
October 5, 2018
Article copyright:
© Copyright 2018
American Mathematical Society
