Properties of integrals with respect to a general stochastic measure in a stochastic heat equation
Author:
V. M. Radchenko
Translated by:
Oleg Klesov
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 |
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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.
References
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- 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
- V. N. Radchenko, Integrals with Respect to Stochastic Measures, Proceedings of Institute of Mathematics, Academy of Sciences of Ukraine, Kiev, vol. 27, 1999. (Russian)
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- 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, DOI https://doi.org/10.1090/S0094-9000-05-00639-3
- 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, DOI https://doi.org/10.1090/S0094-9000-08-00722-9
- 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, DOI https://doi.org/10.1007/s11253-009-0184-2
- Vadym Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554, DOI https://doi.org/10.4064/sm194-3-2
- 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
- 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, DOI https://doi.org/10.1137/S0040585X97984048
- 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, DOI https://doi.org/10.1016/j.anihpb.2005.06.003
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
References
- J. Memin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Stat. Probab. Letters 27 (2001), no. 2, 197–206. MR 1822771 (2002b:60096)
- S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198 (94k:60074)
- V. N. Radchenko, Integrals with Respect to Stochastic Measures, Proceedings of Institute of Mathematics, Academy of Sciences of Ukraine, Kiev, vol. 27, 1999. (Russian)
- V. N. Radchenko, Integrals with respect to stochastic measures and random linear functionals, Teor. Veroyatnost. i Primenen. 36 (1991), no. 3, 594–596; English transl. in Theory Probab. Appl. 36 (1991), no. 3, 621–623. MR 1141138 (93e:60093)
- V. M. Radchenko, On the product of random and real measures, Teor. Imovir. Mat. Stat. 70 (2004), 144–148; English transl. in Theory Probab. Math. Statist. 70 (2005), 161–166. MR 2110872 (2005h:60146)
- V. M. Radchenko, Parameter-dependent integrals with general stochastic measures, Teor. Imovir. Mat. Stat. 75 (2006), 140–143; English transl. in Theory Probab. Math. Statist. 75 (2007), 161–165. MR 2321189 (2008k:60113)
- V. N. Radchenko, The heat equation and the wave equation with general stochastic measures, Ukrain. Mat. Zh. 60 (2008), no. 12, 1675–1685; English transl. in Ukrainian Math. J. 60 (2008), no. 12, 1968–1981. MR 2523115 (2010d:60117)
- V. M. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554 (2010j:60157)
- V. N. Radchenko, On a definition of the integral of a random function, Teor. Veroyatnost. i Primenen. 41 (1996), no. 3, 677–682; English transl. in Theory Probab. Appl. 41 (1996), no. 3, 597–601. MR 1450086 (98f:60002)
- V. N. Radchenko, Sample functions of stochastic measures and Besov spaces, Teor. Veroyatnost. i Primenen. 54 (2009), no. 1, 158–166; English transl. in Theory Probab. Appl. 54 (2010), 160–168. MR 2766653
- S. Kwapień, M. Marcus, and J. Rosiński, Two results on continuity and boundedness of stochastic convolutions, Annales de l’Institut Henri Poincaré (B) Probability and Statistics 42 (2006), no. 5, 553–566. MR 2259974 (2007j:60059)
- A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836 (31:6062)
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Additional Information
V. M. Radchenko
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
vradchenko@univ.kiev.ua
Keywords:
Random measure,
stochastic integral,
stochastic Fubini theorem,
stochastic convolution
Received by editor(s):
October 5, 2009
Published electronically:
August 4, 2011
Additional Notes:
The research is supported by the Swedish Institute, grant SI-01424/2007
Article copyright:
© Copyright 2011
American Mathematical Society