Cable equation with a general stochastic measure

Radchenko, V.

doi:10.1090/S0094-9000-2012-00856-9

Cable equation with a general stochastic measure

Author: V. M. Radchenko
Translated by: O. I. Klesov
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: 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.

References

John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI https://doi.org/10.1007/BFb0074920

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.

Paul C. Bressloff, Cable theory of protein receptor trafficking in a dendritic tree, Phys. Rev. E (3) 79 (2009), no. 4, 041904, 16. MR 2551217, DOI https://doi.org/10.1103/PhysRevE.79.041904

Nathalie Eisenbaum, Mohammud Foondun, and Davar Khoshnevisan, Dynkin’s isomorphism theorem and the stochastic heat equation, Potential Anal. 34 (2011), no. 3, 243–260. MR 2782972, DOI https://doi.org/10.1007/s11118-010-9193-x

Gopinath Kallianpur and Jie Xiong, Stochastic differential equations in infinite-dimensional spaces, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 26, Institute of Mathematical Statistics, Hayward, CA, 1995. Expanded version of the lectures delivered as part of the 1993 Barrett Lectures at the University of Tennessee, Knoxville, TN, March 25–27, 1993; With a foreword by Balram S. Rajput and Jan Rosinski. MR 1465436

Zhiyuan Huang, Caishi Wang, and Xiangjun Wang, Quantum cable equations in terms of generalized operators, Acta Appl. Math. 63 (2000), no. 1-3, 151–164. Recent developments in infinite-dimensional analysis and quantum probability. MR 1831253, DOI https://doi.org/10.1023/A%3A1010776120045

Vlad Bally, Annie Millet, and Marta Sanz-Solé, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab. 23 (1995), no. 1, 178–222. MR 1330767

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. M. Radchenko, Properties of integrals with respect to a general stochastic measure in a stochastic heat equation, Teor. Ĭmovīr. Mat. Stat. 82 (2010), 104–114 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 82 (2011), 103–114. MR 2790486, DOI https://doi.org/10.1090/S0094-9000-2011-00830-7

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, DOI https://doi.org/10.1016/S0167-7152%2800%2900157-7

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 general stochastic measures, Proceedings of Institute of Mathematics, Academy of Sciences of Ukraine 27 (1999). (Russian)

Anna Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63–77. MR 1750304