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Theory of Probability and Mathematical Statistics

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Cable equation with a general stochastic measure


Author: V. M. Radchenko
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 84 (2011).
Journal: Theor. Probability and Math. Statist. 84 (2012), 131-138
MSC (2010): Primary 60G57, 60H15, 60H05
DOI: https://doi.org/10.1090/S0094-9000-2012-00856-9
Published electronically: August 2, 2012
MathSciNet review: 2857423
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the stochastic cable equation that involves an integral with respect to a general random measure. We prove that the paths of the mild solution of the equation are Hölder continuous.


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  • 1. J. B. Walsh, An introduction to stochastic partial differential equations, Lect. Notes Math., vol. 1180, 1986, pp. 265-439. MR 0876085 (88a:60114)
  • 2. H. Tuckwell, F. Wan, and Y. Wong, The interspike interval of a cable model neuron with white noise input, Biol. Cybern. 49 (1984), no. 3, 155-167.
  • 3. P. C. Bressloff, Cable theory of protein receptor trafficking in a dendritic tree, Phys. Rev. E 79 (2009), no. 4, 041904-1-041904-16. MR 2551217 (2010k:92017)
  • 4. N. Eisenbaum, M. Foondun, and D. Khoshnevisan, Dynkin's isomorphism theorem and the stochastic heat equation, Potential Anal. 34 (2011), no. 3, 243-260. MR 2782972
  • 5. G. Kallianpur and J. Xiong, Stochastic Differential Equations in Infinite-Dimensional Spaces, IMS Lecture Notes-Monograph Series, vol. 26, Institute of Mathematical Statistics, Hayward, CA, 1995. MR 1465436 (98h:60001)
  • 6. Z. Huang, C. Wang, and X. Wang, Quantum cable equations in terms of generalized operators, Acta Appl. Math. 63 (2000), no. 1-3, 151-164. MR 1831253 (2002b:81071)
  • 7. V. Bally, A. Millet, and M. Sanz-Solè, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab. 23 (1995), no. 1, 178-222. MR 1330767 (96d:60091)
  • 8. V. N. Radchenko, Heat equation and wave equation with general stochastic measures, Ukr. Mat. Zh. 60 (2008), no. 12, 1675-1685; English transl. in Ukr. Math. J. 60 (2008), no. 12, 1968-1981. MR 2523115 (2010d:60117)
  • 9. 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)
  • 10. V. M. Radchenko, Properties of integrals with respect to a general stochastic measure in a stochastic heat equation, Teor. Imovirnost. Matem. Statist. 82 (2010), 104-114; English transl. in Theor. Probability Math. Statist. 82 (2011), 103-114. MR 2790486 (2011m:60166)
  • 11. J. Memin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Stat. Probab. Lett. 27 (2001), no. 2, 197-206. MR 1822771 (2002b:60096)
  • 12. S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198 (94k:60074)
  • 13. V. N. Radchenko, Integrals with respect to general stochastic measures, Proceedings of Institute of Mathematics, Academy of Sciences of Ukraine 27 (1999). (Russian)
  • 14. A. Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63-77. MR 1750304 (2001e:46058)

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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 4E, Kiev 03127, Ukraine
Email: vradchenko@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2012-00856-9
Keywords: Stochastic measure, stochastic partial differential equation, stochastic cable equation, mild solution
Received by editor(s): March 17, 2011
Published electronically: August 2, 2012
Article copyright: © Copyright 2012 American Mathematical Society

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