Approximation of solutions of the wave equation driven by a stochastic measure
Authors:
V. M. Radchenko and N. O. Stefans’ka
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 99 (2019), 229-238
MSC (2010):
Primary 60H15; Secondary 60H05, 60G57
DOI:
https://doi.org/10.1090/tpms/1092
Published electronically:
February 27, 2020
MathSciNet review:
3908668
Full-text PDF
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Additional Information
Abstract: The mild solution of the wave equation driven by a general stochastic measure is considered. It is proved that solutions of this equation converge if paths of stochastic measures converge.
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), 1–16. MR 3553450
- I. Bodnarchuk, Mild solution of the wave equation with a general random measure, Visnyk Kyiv University. Mathematics. Mechanics 24 (2010), 28–33. (Ukrainian)
- 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 stable noise, Teor. Imovirnost. Matem. Statyst. 96 (2017), 142–154; English transl. in Theory Probab. Math. Statist. 96 (2018), 145–157. MR 3666878
- F. J. Delgado-Vences and M. Sanz-Solé, Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm, Bernoulli 20 (2014), 2169–2216. MR 3263102
- F. J. Delgado-Vences and M. Sanz-Solé, Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22 (2016), 1572–1597. MR 3474826
- V. M. Radchenko and N. O. Stefans’ka, Fourier and Fourier-Haar series for stochastic measures, Teor. Imovirnost. Matemem. Statyst. 96 (2017), 155–162; English transl. in Theory Probab. Math. Statist. 96 (2018), 159–167. MR 3666879
- V. M. Radchenko and N. O. Stefans’ka, Fourier transform of general stochastic measures, Teor. Imovirnost. Matemem. Statyst. 94 (2016), 143–149; English transl. in Theory Probab. Math. Statist. 94 (2017), 151–158. MR 3553460
- G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, Chapman and Hall, London, 1994. MR 1280932
- S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992.
- V. N. Radchenko, Integrals with respect to general stochastic measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
- M. Talagrand, Les mesures vectorielles a valeurs dans $L_0$ sont bornées, Ann. Sci. École Norm. Sup. 14 (1981), 445–452. MR 654206
- V. M. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Teor. Veroyatnost. Primenen. 59 (2015), no. 2, 375–386; English transl. in Theory Probab. Appl. 59 (2015), 328–339. MR 3416054
- V. N. Radchenko, Sample functions of stochastic measures and Besov spaces, Teor. Veroyatnost. Primenen. 54 (2010), no. 1, 161–169; English transl. in Theory Probab. Appl. 54 (2010), 160–168. MR 2766653
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanian, Probability Distributions on Banach Spaces, D. Reidel Publishing Co., Dordrecht, 1987. MR 1435288
- B. S. Kashin and A. A. Saakyan, Orthogonal series, “Nauka”, Moscow, 1984; English transl., AMS, Providence, 1989. MR 779286
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Additional Information
V. M. Radchenko
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
vradchenko@univ.kiev.ua
N. O. Stefans’ka
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
neliastefanska@gmail.com
Keywords:
Stochastic measure,
stochastic wave equation,
mild solution,
Fourier–Haar series
Received by editor(s):
April 2, 2018
Published electronically:
February 27, 2020
Article copyright:
© Copyright 2020
American Mathematical Society