Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
Equation for vibrations of a fixed string driven by a general stochastic measure
I. M. Bodnarchuk, V. M. Radchenko
Abstract: Equation for vibrations of a string with fixed ends driven by a general stochastic measure is investigated in three cases: the stochastic measure depends on time variable, on space variable and on the set of both variables. Averaging principle is considered and the rate of convergence to the solution of the averaged equation is evaluated.
1. L. I. Rusaniuk, G. M. Shevchenko, Wave equation for a homogeneous string with fixed ends driven by a stable random noise, Theory Probab. Math. Statist., 98 (2019), 171-181.
2. E. Orsingher, Randomly forced vibrations of a string, Annales de l'I. H. P., section B, 18 (1982), no. 4, 367-394.
3. D. Khoshnevisan, E. Nualart, Level sets of the stochastic wave equationdriven by a symmetric L'evy noise, Bernoulli, 14 (2008), no. 4, 899-925.
4. L. Pryhara, G. Shevchenko, Wave equation with a coloured stable noise, Random Operators and Stochastic Equations, 25 (2017), no. 4, 249-260.
5. L. I. Rusaniuk, G. M. Shevchenko, Wave equation with stable noise, Theory Probab. Math. Statist., 96 (2018), no. 1, 145-157.
6. I. M. Bodnarchuk, Wave equation with a stochastic measure, Theory Probab. Math. Statist., 94 (2017), 1-16.
7. I. M. Bodnarchuk, V. M. Radchenko Wave equation in a plane driven by a general stochastic measure, Theory Probab. Math. Statist., 98 (2019), 73-90.
8. I. M. Bodnarchuk, V. M. Radchenko Wave equation in three-dimensional space driven by a general stochastic measure, Teor. Imovir. Matem. Statist., 100 (2019), 43-59. (Ukrainian)
9. J. Duan, W.Wang, Effective Dynamics of Stochastic Partial Dierential Equations, Birkhauser, Boston, 1992.
10. P. Gao, Averaging principle for stochastic Korteweg-de Vries equation, J. Diferential Equations, In Press, https://doi.org/10.1016/j.jde.2019.07.012
11. V. M. Radchenko Averaging principle for heat equation driven by general stochastic measure, Statist. Probab. Lett., 146 (2019), 224-230.
12. V. Radchenko, Averaging principle for equation driven by a stochastic measure, Stochastics, 91 (2019), no. 6, 905-915.
13. S. Kwapien, W. A. Woyczynski, Random Series and Stochastic Integrals: Single and Multiple, Birkhauser, Boston, 1992.
14. Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Topics, Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008.
15. C. Tudor, On the Wiener integral with respect to a sub-fractional Brownian motion on an interval, J. Math. Anal. Appl., 351 (2009), 456-468.
16. L. Drewnowski, Topological rings of sets, continuous set functions, integration. III, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 20 (1972), 439-445.
17. V. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math., 194 (2009), no. 3, 231-251.
18. G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Innite Variance, Chapmen & Hall, Boca Raton, 1994.
19. V. N. Radchenko Sample functions of stochastic measures and Besov spaces, Theory Probab. Appl., 54 (2010), no. 1, 160-168.
20. P. L. Chow, Stochastic partial dierential equations, Chapman and Hall/CRC, 2014.
21. H. Fu, L. Wan, J. Liu, Strong convergence in averaging principle for stochastic hyperbolic parabolic equations with two time-scales, Stoch. Proc. Appl., 125 (2015), no. 8, 3255-3279.
22. A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 2002.
23. N. K. Bary, A Treatise on Trigonometric Series. Vol. 1, Pergamon Press, OxfordNew York, 1964.
24. V. M. Radchenko, N. O. Stefans'ka, Fourier and Fourier-Haar series for stochastic measures, Theory Probab. Math. Statist., 96 (2018), 159-167.