Properties of integrals with respect to a general stochastic measure in a stochastic heat equation

Radchenko, V.

doi:10.1090/S0094-9000-2011-00830-7

Properties of integrals with respect to a general stochastic measure in a stochastic heat equation

Author: V. M. Radchenko
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 82 (2010).
Journal: Theor. Probability and Math. Statist. 82 (2011), 103-114
MSC (2010): Primary 60G57, 60H15, 60H05
DOI: https://doi.org/10.1090/S0094-9000-2011-00830-7
Published electronically: August 4, 2011
MathSciNet review: 2790486
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a theorem on the continuity with respect to a parameter and an analogue of Fubini's theorem for integrals with respect to a general stochastic measure defined on Borel subsets of $\mathbb{R}$ . These results are applied to study the stochastic heat equation considered in a mild as well as in a weak form.

1. Jean Mémin, Yulia Mishura, and Esko Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), no. 2, 197–206. MR 1822771, https://doi.org/10.1016/S0167-7152(00)00157-7
2. 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
3. V. N. Radchenko, Integrals with Respect to Stochastic Measures, Proceedings of Institute of Mathematics, Academy of Sciences of Ukraine, Kiev, vol. 27, 1999. (Russian)
4. V. N. Radchenko, Integrals with respect to random measures and random linear functionals, Teor. Veroyatnost. i Primenen. 36 (1991), no. 3, 594–596 (Russian); English transl., Theory Probab. Appl. 36 (1991), no. 3, 621–623 (1992). MR 1141138, https://doi.org/10.1137/1136077
5. V. M. Radchenko, On the product of random and real measures, Teor. Ĭmovīr. Mat. Stat. 70 (2004), 144–148 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 70 (2005), 161–166. MR 2110872, https://doi.org/10.1090/S0094-9000-05-00639-3
6. V. M. Radchenko, Parameter-dependent integrals with general random measures, Teor. Ĭmovīr. Mat. Stat. 75 (2006), 140–143 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 161–165. MR 2321189, https://doi.org/10.1090/S0094-9000-08-00722-9
7. 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, https://doi.org/10.1007/s11253-009-0184-2
8. Vadym Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554, https://doi.org/10.4064/sm194-3-2
9. V. N. Radchenko, On a definition of the integral of a random function, Teor. Veroyatnost. i Primenen. 41 (1996), no. 3, 677–682 (Russian, with Russian summary); English transl., Theory Probab. Appl. 41 (1996), no. 3, 597–601 (1997). MR 1450086
10. V. N. Radchenko, Sample functions for random measures and Besov spaces, Teor. Veroyatn. Primen. 54 (2009), no. 1, 158–166 (Russian, with Russian summary); English transl., Theory Probab. Appl. 54 (2010), no. 1, 160–168. MR 2766653, https://doi.org/10.1137/S0040585X97984048
11. Stanisław Kwapień, Michael B. Marcus, and Jan Rosiński, Two results on continuity and boundedness of stochastic convolutions, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 5, 553–566 (English, with English and French summaries). MR 2259974, https://doi.org/10.1016/j.anihpb.2005.06.003
12. Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836