Theory of Probability and Mathematical Statistics
Equation for vibrations of a string with fixed ends, forced by a stable random noise
L. I. Rusaniuk, G. M. Shevchenko
Download PDF
Abstract: We study the equation for forced vibrations of a homogeneous string with a random force having a symmetric α-stable distribution. We show that the function constructed by the Fourier method is a generalized solution to the equation, and establish its pathwise regularity.
Keywords: Equation for vibrations of a string, wave equation, Fourier method, generalized solution, stable measure with independent increments, LePage series.
Bibliography: 1. I. M. Bodnarchuk, Wave equation with a stochastic measure, Theory Probab. Math. Statist., 94 (2016), 1-15.
2. R. Carmona, D. Nualart, Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Related Fields, 79 (1988), no. 4, 469-508.
3. E. Cabana, The vibrating string forced by white noise, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 111-130.
4. R. C. Dalang, N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab., 26 (1998), no. 1, 187-212.
5. R. C. Dalang, M. Sanz-Sole, Holder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., 199(931), AMS, Providence, 2009.
6. 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 (2012), 53-59.
7. N. Kono, M. Maejima, Holder continuity of sample paths of some self-similar stable processes, Tokyo J. Math., 14 (1991), no. 1, 93-100.
8. D. Khoshnevisan, E. Nualart, Level sets of the stochastic wave equation driven by a symmetric Levy noise, Bernoulli, 14 (2008), no. 4, 899-925.
9. M. A. Lifshits, Gaussian random functions, Kluwer Academic Publishers, Dordrecht, 1995.
10. S. V. Lototsky, B. L. Rozovsky, Stochastic partial differential equations, Universitext, Springer, Cham, 2017.
11. A. Millet, P.-L. Morien, On a stochastic wave equation in two space dimensions: regularity of the solution and its density, Stochastic Processes Appl., 86 (2000), no. 1, 141-162.
12. B. Oksendal, F. Proske, M. Signahl, The Cauchy problem for the wave equation with Levy noise initial data, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), no. 2, 249-270.
13. E. Orsingher, Randomly forced vibrations of a string, Ann. Inst. H. Poincare Sect. B (N.S.), 18 (1982), no. 4, 367-394.
14. L. Pryhara, G. Shevchenko, Stochastic wave equation in a plane driven by spatial stable noise, Mod. Stoch. Theory Appl., 3 (2016), no. 3, 237-248.
15. L. Pryhara, G. Shevchenko, Wave equation with a stable noise, Theory Probab. Math. Statist., 96 2017, 143-155.
16. L. Pryhara, G. Shevchenko, Wave equation with a coloured stable noise, Random Oper. Stoch. Equ., 25 (2017), no. 4, 249-260.
17. L. Quer-Sardanyons, S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Processes Appl., 177 (2007), no. 10, 1448-1472.
18. G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994.
19. 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, 265-439.
