Heat equation and wave equation with general stochastic measures
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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 sigma-additivity 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.
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- 1.V. I. Klyatskin, Stochastic Equations through the Eye of the Physicist [in Russian], Fizmatlit, Moscow (2001).Google Scholar
- 3.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).Google Scholar
- 5.B. L. Rozovskii, Evolution Stochastic Systems [in Russian], Nauka, Moscow (1983).Google Scholar
- 7.J. B. Walsh, “An introduction to stochastic partial differential equations,” Lect. Notes Math., 1180, 236–434 (1984).Google Scholar
- 11.H. Holden, B. Óksendal, L. Ubóe, and T. Zhang, Stochastic Partial Differential Equations. A Modelling White Noise Functional Approach, Birkhäuser, Boston (1996).Google Scholar
- 15.V. N. Radchenko, Integrals with Respect to General Random Measures [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1999).Google Scholar
- 17.V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1971).Google Scholar