Wave equation with a stochastic measure
Author:
I. M. Bodnarchuk
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 94 (2017), 1-16
MSC (2010):
Primary 60H15; Secondary 60G17, 60G57
DOI:
https://doi.org/10.1090/tpms/1005
Published electronically:
August 25, 2017
MathSciNet review:
3553450
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Additional Information
Abstract: The Cauchy problem for the wave equation on the line driven by a general stochastic measure is studied. The existence, uniqueness, and Hölder regularity of the mild solution are proved. The continuous dependence of the solution on the data is established.
References
- Stanisław Kwapień and Wojbor A. Woyczyński, Random series and stochastic integrals: single and multiple, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1167198
- I. Bodnarchuk, Mild solution of a wave equation with a general random measure, Visnyk Kyiv University. Mathematics and Mechanics 24 (2010), 28–33. (Ukrainian)
- Vadym Radchenko, Heat equation with general stochastic measure colored in time, Mod. Stoch. Theory Appl. 1 (2014), no. 2, 129–138. MR 3316482, DOI https://doi.org/10.15559/14-VMSTA14
- Vadym Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554, DOI https://doi.org/10.4064/sm194-3-2
- Ī. M. Bodnarchuk and G. M. Shevchenko, The heat equation in a multidimensional domain with a general stochastic measure, Teor. Ĭmovīr. Mat. Stat. 93 (2015), 7–21 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 93 (2016), 1–17. MR 3553436, DOI https://doi.org/10.1090/tpms/991
- Raluca M. Balan and Ciprian A. Tudor, The stochastic wave equation with fractional noise: a random field approach, Stochastic Process. Appl. 120 (2010), no. 12, 2468–2494. MR 2728174, DOI https://doi.org/10.1016/j.spa.2010.08.006
- Robert C. Dalang and Marta Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc. 199 (2009), no. 931, vi+70. MR 2512755, DOI https://doi.org/10.1090/memo/0931
- V. N. Radchenko, Integrals with Respect to General Stochastic Measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kiev, 1999. (Russian)
- V. M. Radchenko, Integral equations with a general random measure, Teor. Ĭmovīr. Mat. Stat. 91 (2014), 154–163 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 91 (2015), 169–179. MR 3364132, DOI https://doi.org/10.1090/tpms/975
- V. M. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Theory Probab. Appl. 59 (2015), no. 2, 328–339. MR 3416054, DOI https://doi.org/10.1137/S0040585X97T987119
- Anna Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63–77. MR 1750304
References
- S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198
- I. Bodnarchuk, Mild solution of a wave equation with a general random measure, Visnyk Kyiv University. Mathematics and Mechanics 24 (2010), 28–33. (Ukrainian)
- V. Radchenko, Heat equation with general stochastic measure colored in time, Modern Stochastics: Theory and Applications 1 (2014), 129–138. MR 3316482
- V. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554
- I. M. Bodnarchuk and G. M. Shevchenko, Heat equation in a multidimensional domain with a general stochastic measure, Teor. Imovirnost. Matem. Statyst. 93 (2015), 7–21; English transl. in Theor. Probability and Math. Statist. 93 (2016), 19–31. MR 3553436
- 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
- R. C. Dalang and M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Memoirs of the American Mathematical Society 199 (2009), no. 931. MR 2512755
- V. N. Radchenko, Integrals with Respect to General Stochastic Measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kiev, 1999. (Russian)
- V. M. Radchenko, Integral equations with a general stochastic measure, Teor. Imovirnost. Matem. Statyst. 91 (2014), 154–163; English transl. in Theor. Probability and Math. Statist. 91 (2015), 169–179. MR 3364132
- V. N. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Teor. Veroyatnost. Primenen. 59 (2014), no. 2, 375–386; English transl. in Theory Probab. Appl. 59 (2015), no. 2, 328–339. MR 3416054
- A. Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. 13 (1997), no. 2, 63–77. MR 1750304
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Additional Information
I. M. Bodnarchuk
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
robeiko_i@ukr.net
Keywords:
Stochastic measure,
stochastic wave equation,
mild solution,
Hölder condition,
Besov space
Received by editor(s):
April 5, 2016
Published electronically:
August 25, 2017
Article copyright:
© Copyright 2017
American Mathematical Society
