A Journal "Theory of Probability and Mathematical Statistics"
2024
2023
2022
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process   Special Issues   History  

Theory of Probability and Mathematical Statistics



Approximation of solution of the cable equation driven by a stochastic measure

B. I. Manikin, V. M. Radchenko

Link

Abstract: The cable equation driven by a general stochastic measure is studied. We prove that convergence of the sequence of stochastic integrators implies the convergence of the sequence of solutions. Fourier series approximation of integrator is also considered.

Keywords: Stochastic measure, stochastic cable equation, mild solution

Bibliography:
1. P. C. Bressloff, Waves in neural media: From single neurons to neural fields, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, New York, 2014.
2. G. Kallianpur, Stochastic differential equation models for spatially distributed neurons and propagation of chaos for interacting systems, Math. Biosci. 112 (1992), no. 2, 207 – 224.
3. G. Kallianpur and J. Xiong, Stochastic differential equations in infinite-dimensional spaces, Lecture notes-Monograph series, vol. 26, Institute of Mathematical Statistics, Hayward, 1995.
4. C. Koch, Biophysics of computation: Information processing in single neurons, Computational Neuroscience, Oxford University Press, New York, 1999.
5. S. Kwapie\'{n} and W. A. Woyczy\'{n}ski, Random series and stochastic integrals: Single and multiple, Birkh\"{a}user, Boston, 1992.
6. B. Manikin and V. Radchenko, Approximation of the solution to the parabolic equation driven by stochastic measure, Theory Probab. Math. Statist. 102 (2020), in press.
7. V. Radchenko, Integrals with respect to general stochastic measures, Institute of Mathematics, Kyiv, 1999, in Russian.
8. V. Radchenko, On the product of a random and a real measure, Theory Probab. Math. Statist. 70(2005), 197–206.
9. V. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Theory Probab. Appl. 59 (2015), 328–339.
10. V. Radchenko, Averaging principle for equation driven by a stochastic measure, Stochastics 91 (2019), no. 6, 905–915.
11. V. Radchenko and N. Stefans'ka, Approximation of solutions of the stochastic wave equation by using the Fourier series, Modern Stochastics: Theory and Applications 5 (2018), no. 4, 429–444.
12. V. Radchenko and N. Stefans'ka, Fourier series and Fourier–Haar series for stochastic measures, Theor. Probability and Math. Statist. (2018), no. 96, 159–167.
13. V. M. Radchenko, Cable equation with a general stochastic measure, Theor. Probability and Math. Statist. (2012), no. 84, 131–138.
14. G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian random processes, Chapman and Hall, London, 1994.
15. M. Talagrand, Les mesures vectorielles \`{a} valeurs dans ${L}_0$ sont born\'{e}es, Annales scientifiques de l'\'{E}cole Normale Sup\'{e}rieure 14 (1981), no. 4, 445–452.
16. N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanian, Probability distributions on Banach spaces, D. Reidel Publishing Co., Dordrecht, 1987.
17. J. B. Walsh, An introduction to stochastic partial differential equations, Lect. Notes Math. 1180 (1986), 265–439.
18. A. Zygmund, Trigonometric series. 3rd ed., Cambridge Univ. Press, Cambridge, 2002.