Find out how to access previewonly content
Heat equation and wave equation with general stochastic measures
 V. N. Radchenko
 … show all 1 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
We consider the heat equation and wave equation with constant coefficients that contain a term given by an integral with respect to a random measure. Only the condition of sigmaadditivity in probability is imposed on the random measure. Solutions of these equations are presented. For each equation, we prove that its solutions coincide under certain additional conditions.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1675 – 1685, December, 2008.
 V. I. Klyatskin, Stochastic Equations through the Eye of the Physicist [in Russian], Fizmatlit, Moscow (2001).
 A. Sturm, “On convergence of population processes in random environments to the stochastic heat equation with colored noise,” Electron. J. Probab., 8, No. 6, 1–39 (2003).
 I. M. Gel’fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4, Some Application of Harmonic Analysis. Rigged Hilbert Spaces [in Russian], Fizmatgiz, Moscow (1961).
 E. Pardoux, “Stochastic partial differential equations. A review,” Bull. Sci. Math., Sér. 2 ^{ e }, 117, 29–47 (1993).
 B. L. Rozovskii, Evolution Stochastic Systems [in Russian], Nauka, Moscow (1983).
 K.H. Kim, “Stochastic partial differential equations with variable coefficients in C ^{1} domains,” Stochast. Process. Appl., 112, 261–283 (2004). CrossRef
 J. B. Walsh, “An introduction to stochastic partial differential equations,” Lect. Notes Math., 1180, 236–434 (1984).
 R. C. Dalang, “Extending martingale measure stochastic integral with applications to spatially homogeneous SPDE’s,” Electron. J. Probab., 4, No. 6, 1–29 (1999).
 R. C. Dalang and C. Mueller, “Some nonlinear SPDE’s that are second order in time,” Electron. J. Probab., 8, No. 1, 1–21 (2003).
 D. Conus and R. C. Dalang, “The nonlinear stochastic wave equation in high dimensions,” Electron. J. Probab., 13, No. 22, 629–670 (2008).
 H. Holden, B. Óksendal, L. Ubóe, and T. Zhang, Stochastic Partial Differential Equations. A Modelling White Noise Functional Approach, Birkhäuser, Boston (1996).
 Yu. A. Rozanov, Random Fields and Stochastic Partial Differential Equations [in Russian], Fizmatlit, Moscow (1995).
 J. Memin, Yu. Mishura, and E. Valkeila, “Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion,” Statist. Probab. Lett., 51, 197–206 (2001). CrossRef
 S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston (1992).
 V. N. Radchenko, Integrals with Respect to General Random Measures [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1999).
 V. N. Radchenko, “Integrals with respect to random measures and random linear functionals,” Teor. Ver. Primen., 36, No. 3, 594–596 (1991).
 V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1971).
 A. A. Kirillov and A. D. Gvishiani, Theorems and Problems of Functional Analysis [in Russian], Nauka, Moscow (1979).
 K. Yosida, Functional Analysis, Springer, Berlin (1965).
 V. N. Radchenko, “On convergence of integrals with respect to L _{0}valued measures,” Mat. Zametki, 53, No. 5, 102–106 (1993).
 Title
 Heat equation and wave equation with general stochastic measures
 Journal

Ukrainian Mathematical Journal
Volume 60, Issue 12 , pp 19681981
 Cover Date
 20081201
 DOI
 10.1007/s1125300901842
 Print ISSN
 00415995
 Online ISSN
 15739376
 Publisher
 Springer US
 Additional Links
 Topics
 Industry Sectors
 Authors

 V. N. Radchenko ^{(1)}
 Author Affiliations

 1. Shevchenko Kyiv National University, Kyiv, Ukraine