Wave equation for a homogeneous string with fixed ends driven by a stable random noise
Authors:
L. I. Rusanyuk and G. M. Shevchenko
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 98 (2019), 171-181
MSC (2010):
Primary 60H15, 35L05; Secondary 35R60, 60G52
DOI:
https://doi.org/10.1090/tpms/1069
Published electronically:
August 19, 2019
MathSciNet review:
3824685
Full-text PDF
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Additional Information
Abstract: A wave equation with external forces is considered in this paper for a homogeneous string with fixed ends. The distribution of the right-hand side of the equation is symmetric $\alpha$-stable. It is proved that the function constructed by the Fourier method is a generalized solution of the equation. The regularity of the trajectories is also established.
References
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References
- I. M. Bodnarchuk, Wave equation with a stochastic measure, Teor. Imovirnost. Matem. Statyst. 94 (2016), 1–15; English transl. in Theory Probab. Math. Statist. 94 (2017), 1–15. MR 3553450
- R. Carmona and D. Nualart, Random nonlinear wave equations: smoothness of the solutions, Probab. Theory Related Fields 79 (1988), no. 4, 469–508. MR 966173
- E. Cabaña, The vibrating string forced by white noise, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 15 (1970), 111–130. MR 0279909
- 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. 199 (931), AMS, Providence, 2009. MR 2512755
- D. M. Gorodnya, On the existence and uniqueness of solutions of the Cauchy problem for wave equations with general stochastic measures, Teor. Imovirnost. Matem. Statyst. 85 (2011), 50–55; English transl. in Theory Probab. 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
- D. Khoshnevisan and E. Nualart, Level sets of the stochastic wave equation driven by a symmetric Lévy noise, Bernoulli 14 (2008), no. 4, 899–925. MR 2543579
- M. A. Lifshits, Gaussian Random Functions, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736
- S. V. Lototsky and B. L. Rozovsky, Stochastic Partial Differential Equations, Universitext, Springer, Cham, 2017. MR 3674586
- A. Millet and P.-L. Morien, On a stochastic wave equation in two space dimensions: regularity of the solution and its density, Stoch. Process. Appl. 86 (2000), no. 1, 141–162. MR 1741200
- B. Øksendal, F. Proske, and M. Signahl, The Cauchy problem for the wave equation with Lévy noise initial data, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 2, 249–270. MR 2235547
- E. Orsingher, Randomly forced vibrations of a string, Ann. Inst. H. Poincaré Sect. B (N.S.) 18 (1982), no. 4, 367–394. MR 683337
- L. Pryhara and G. Shevchenko, Stochastic wave equation in a plane driven by spatial stable noise, Modern Stoch. Theory Appl. 3 (2016), no. 3, 237–248. MR 3576308
- L. Pryhara and G. Shevchenko, Wave equation with a stable noise, Teor. Imovirnost. Matem. Statyst. 96 (2017), 142–154; English transl. in Theory Probab. Math. Statist. 96 2017, 143–155. MR 3666878
- L. Pryhara and G. Shevchenko, Wave equation with a coloured stable noise, Random Oper. Stoch. Equ. 25 (2017), no. 4, 249–260. MR 3731389
- L. Quer-Sardanyons and S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stoch. Process. Appl. 177 (2007), no. 10, 1448–1472. MR 2353035
- G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994. MR 1280932
- J. B. Walsh, An introduction to stochastic partial differential equations, Ecole D’ete de Probabilites de Saint-Flour, XIV-1984, Lecture Notes in Math., Springer, Berlin, 1986, pp. 265–439. MR 876085
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Additional Information
L. I. Rusanyuk
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
pruhara7@gmail.com
G. M. Shevchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Wave equation for a string,
wave equation,
Fourier method,
generalized solution,
stable measure with independent increments,
LePage representation
Received by editor(s):
January 6, 2018
Published electronically:
August 19, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
