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Riemann integral of a random function and the parabolic equation with a general stochastic measure


Author: V. Radchenko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 87 (2012).
Journal: Theor. Probability and Math. Statist. 87 (2013), 185-198
MSC (2000): Primary 60H05, 60H15
DOI: https://doi.org/10.1090/S0094-9000-2014-00912-6
Published electronically: March 21, 2014
MathSciNet review: 3241455
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Abstract: For a stochastic parabolic equation driven by a general stochastic measure, the weak solution is obtained. The integral of a random function in the equation is considered as a limit in probability of Riemann integral sums. Basic properties of such integrals are studied in this paper.


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  • 1. S. Albeverio, J.-L. Wua, and T.-S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl. 74 (1998), 21-36. MR 1624076 (99c:60124)
  • 2. G. P. Curbera and O. Delgado, Optimal domains for $ L_0$-valued operators via stochastic measures, Positivity 11 (2007), 399-416. MR 2336205 (2008g:46063)
  • 3. R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e's, Electron. J. Probab. 4 (1999), 1-29. MR 1684157 (2000b:60132)
  • 4. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge Univ. Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • 5. A.  M. Ilyin, A. S. Kalashnikov, and O. A. Oleynik, Linear second-order partial differential equations of the parabolic type, J. Math. Sci. (N. Y.) 108 (2002), 435-542. MR 1875963 (2003a:35087)
  • 6. P. Kotelenez, Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations, Stochastic Modelling Appl. Probab., vol. 58, Springer-Verlag, Berlin-Heidelberg-New York, 2007. MR 2370567 (2009h:60004)
  • 7. S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198 (94k:60074)
  • 8. J. Memin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statistics and Probability Letters 27 (2001), 197-206. MR 1822771 (2002b:60096)
  • 9. S. M. Nikolsky, A Course of Mathematical Analysis, vol. 2, ``Mir'', Moscow, 1977. MR 0466436 (57:6315b)
  • 10. S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204. MR 1486552 (99k:60166)
  • 11. S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise: an evolution equation approach, Encyclopedia Math. Appl., vol. 113, Cambridge Univ. Press, Cambridge, 2007. MR 2356959 (2009b:60200)
  • 12. V. Radchenko, Integrals with Respect to General Stochastic Measures, Institute of Mathematics, Kyiv, 1999. (Russian)
  • 13. V. Radchenko, Heat equation and wave equation with general stochastic measures, Ukrain. Mat. Zh. 60 (2008), 1675-1685; English transl. in Ukrainian Math. J. 60 (2008), 1968-1981. MR 2523115 (2010d:60117)
  • 14. V. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), 231-251. MR 2539554 (2010j:60157)
  • 15. S. Rolewicz, Metric Linear Spaces, Monografie Matematyczne, vol. 56, PWN--Polish Scientific Publishers, Warsaw, 1972. MR 0438074 (55:10993)
  • 16. C. Ryll-Nardzewski and W. A. Woyczyński, Bounded multiplier convergence in measure of random vector series, Proc. Amer. Math. Soc. 53 (1975), 96-98. MR 0385960 (52:6819)
  • 17. M. Talagrand, Les mesures vectorielles à valeurs dans $ L_0$ sont bornées, Ann. Sci. École Norm. Sup. (4) 14 (1981), 445-452. MR 654206 (83f:28007)
  • 18. Ph. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976). MR 0423044 (54:11028)
  • 19. J. B. Walsh, An introduction to stochastic partial differential equations, Lect. Not. Math. 1180 (1984), 236-434.

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Additional Information

V. Radchenko
Affiliation: Department of Mathematical Analysis, Kyiv National Taras Shevchenko University, Kyiv 01601, Ukraine
Email: vradchenko@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2014-00912-6
Keywords: Stochastic measure, Riemann integral, stochastic parabolic equation, weak solution
Received by editor(s): December 22, 2011
Published electronically: March 21, 2014
Additional Notes: This research was supported by Alexander von Humboldt Foundation, grant no. UKR/1074615. The author wishes to thank Professor M. Zähle for fruitful discussions, and the hospitality of Jena University is gratefully acknowledged
Article copyright: © Copyright 2014 American Mathematical Society

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