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), 5359
MSC (2010):
Primary 60G57, 60H15, 60H05, 35L05, 46F99
Published electronically:
January 11, 2013
MathSciNet review:
2933702
Fulltext PDF
Abstract 
References 
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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.
 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), 922951.
 2.
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 pp.\ (electronic). MR 1684157
(2000b:60132), 10.1214/EJP.v443
 3.
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
(99c:60127), 10.1214/aop/1022855416
 4.
Lluís
QuerSardanyons and Samy
Tindel, The 1d stochastic wave equation driven by a fractional
Brownian sheet, Stochastic Process. Appl. 117 (2007),
no. 10, 1448–1472. MR 2353035
(2008j:60152), 10.1016/j.spa.2007.01.009
 5.
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
(2010d:60117), 10.1007/s1125300901842
 6.
V.
S. Vladimirov, Uravneniya matematicheskoi fiziki, 4th ed.,
“Nauka”, Moscow, 1981 (Russian). 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), 2833. (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.
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
(94k:60074)
 10.
K. Yosida, Functional Analysis, Springer, Berlin, 1965.
 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), 922951.
 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, 129. 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, 187212. MR 1617046 (99c:60127)
 4.
 L. QuerSardanyons and S. Tindel, The 1d stochastic wave equation driven by a fractional Brownian sheet, Stoch. Prosses. Appl. 17 (2007), 14481472. MR 2353035 (2008j:60152)
 5.
 V. N. Radchenko, Heat equation and wave equation with general stochastic measures, Ukr. Matem. Zh. 60 (2008), no. 12, 16751685; English transl. in Ukrain. Math. J. 60 (2008), 19681981. 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), 2833. (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.
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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:
http://dx.doi.org/10.1090/S009490002013008734
PII:
S 00949000(2013)008734
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
