On the existence and uniqueness of solutions of the Cauchy problem for wave equations with general stochastic measures
Author:
D. M. Gorodnya
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 85 (2012), 53-59
MSC (2010):
Primary 60G57, 60H15, 60H05, 35L05, 46F99
DOI:
https://doi.org/10.1090/S0094-9000-2013-00873-4
Published electronically:
January 11, 2013
MathSciNet review:
2933702
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The existence and uniqueness of a solution of the Cauchy problem for wave equations containing a term expressed via the integral with respect to a stochastic measure are proved. Some general properties of generalized functions assuming values in a Fréchet space are used in the proof of the uniqueness of a solution.
References
- A. Millet and P. Morien, On a non linear stochastic wave equation in the plane: existence and uniqueness of the solution, Ann. Appl. Probab. 11 (2007), 922–951.
- Robert C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29. MR 1684157, DOI https://doi.org/10.1214/EJP.v4-43
- Robert C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212. MR 1617046, DOI https://doi.org/10.1214/aop/1022855416
- Lluís Quer-Sardanyons and Samy Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Process. Appl. 117 (2007), no. 10, 1448–1472. MR 2353035, DOI https://doi.org/10.1016/j.spa.2007.01.009
- V. N. Radchenko, The heat equation and the wave equation with general stochastic measures, Ukraïn. Mat. Zh. 60 (2008), no. 12, 1675–1685 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 60 (2008), no. 12, 1968–1981. MR 2523115, DOI https://doi.org/10.1007/s11253-009-0184-2
- V. S. Vladimirov, Uravneniya matematicheskoĭ fiziki, 4th ed., “Nauka”, Moscow, 1981 (Russian). MR 653331
- I. Bodnarchuk, Mild solution of the wave equation with a general random measure, Visnyk Kyiv Nats. Univer. Mathematics. Mechanics 24 (2010), 28–33. (Ukrainian)
- V. N. Radchenko, Integrals With Respect to General Random Measures, Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
- 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
- K. Yosida, Functional Analysis, Springer, Berlin, 1965.
References
- A. Millet and P. Morien, On a non linear stochastic wave equation in the plane: existence and uniqueness of the solution, Ann. Appl. Probab. 11 (2007), 922–951.
- R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E’s, Electronic J. Probab. 4 (1999), no. 6, 1–29. MR 1684157 (2000b:60132)
- 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 (99c:60127)
- L. Quer-Sardanyons and S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stoch. Prosses. Appl. 17 (2007), 1448–1472. MR 2353035 (2008j:60152)
- V. N. Radchenko, Heat equation and wave equation with general stochastic measures, Ukr. Matem. Zh. 60 (2008), no. 12, 1675–1685; English transl. in Ukrain. Math. J. 60 (2008), 1968–1981. MR 2523115 (2010d:60117)
- V. S. Vladimirov, Equations of Mathematical Physics, “Nauka”, Moscow, 1981; English transl., “Mir”, Moscow, 1979. MR 653331 (83i:00029)
- I. Bodnarchuk, Mild solution of the wave equation with a general random measure, Visnyk Kyiv Nats. Univer. Mathematics. Mechanics 24 (2010), 28–33. (Ukrainian)
- V. N. Radchenko, Integrals With Respect to General Random Measures, Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
- S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198 (94k:60074)
- K. Yosida, Functional Analysis, Springer, Berlin, 1965.
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60G57,
60H15,
60H05,
35L05,
46F99
Retrieve articles in all journals
with MSC (2010):
60G57,
60H15,
60H05,
35L05,
46F99
Additional Information
D. M. Gorodnya
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4e, Kiev 03127, Ukraine
Email:
gorodnyaya@yandex.ru
Keywords:
Stochastic wave equation,
Cauchy problem,
stochastic measure,
Fréchet space,
generalized function
Received by editor(s):
June 23, 2011
Published electronically:
January 11, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
