Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 85 (2011).
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

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 [Enhancements On Off] (What's this?)

  • 1. 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.
  • 2. 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)
  • 3. 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)
  • 4. 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)
  • 5. 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)
  • 6. V. S. Vladimirov, Equations of Mathematical Physics, ``Nauka'', Moscow, 1981; English transl., ``Mir'', Moscow, 1979. MR 653331 (83i:00029)
  • 7. I. Bodnarchuk, Mild solution of the wave equation with a general random measure, Visnyk Kyiv Nats. Univer. Mathematics. Mechanics 24 (2010), 28-33. (Ukrainian)
  • 8. V. N. Radchenko, Integrals With Respect to General Random Measures, Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
  • 9. S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198 (94k:60074)
  • 10. 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

DOI: https://doi.org/10.1090/S0094-9000-2013-00873-4
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

American Mathematical Society