Approximation of solution of the cable equation driven by a stochastic measure
Authors:
B. I. Manikin and V. M. Radchenko
Journal:
Theor. Probability and Math. Statist. 104 (2021), 103-112
MSC (2020):
Primary 60H15, 60G57, 60H05
DOI:
https://doi.org/10.1090/tpms/1148
Published electronically:
September 24, 2021
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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.
References
- Paul C. Bressloff, Waves in neural media, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, New York, 2014. From single neurons to neural fields. MR 3136844, DOI 10.1007/978-1-4614-8866-8
- Gopinath Kallianpur, Stochastic differential equation models for spatially distributed neurons and propagation of chaos for interacting systems, Math. Biosci. 112 (1992), no. 2, 207–224. MR 1196373, DOI 10.1016/0025-5564(92)90024-Q
- 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
- C. Koch, Biophysics of computation: Information processing in single neurons, Computational Neuroscience, Oxford University Press, New York, 1999.
- S. Kwapień and W. A. Woyczyński, Random series and stochastic integrals: Single and multiple, Birkhäuser, Boston, 1992.
- 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.
- V. Radchenko, Integrals with respect to general stochastic measures, Institute of Mathematics, Kyiv, 1999, in Russian.
- —, On the product of a random and a real measure, Theory Probab. Math. Statist. 70 (2005), 197–206.
- V. M. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Theory Probab. Appl. 59 (2015), no. 2, 328–339. MR 3416054, DOI 10.1137/S0040585X97T987119
- V. M. Radchenko, Averaging principle for equation driven by a stochastic measure, Stochastics 91 (2019), no. 6, 905–915. MR 3985803, DOI 10.1080/17442508.2018.1559320
- Vadym Radchenko and Nelia Stefans’ka, Approximation of solutions of the stochastic wave equation by using the Fourier series, Mod. Stoch. Theory Appl. 5 (2018), no. 4, 429–444. MR 3914723, DOI 10.15559/18-vmsta115
- —, Fourier series and Fourier – Haar series for stochastic measures, Theor. Probability and Math. Statist. (2018), no. 96, 159–167.
- V. M. Radchenko, A cable equation with a general stochastic measure, Teor. Ĭmovīr. Mat. Stat. 84 (2011), 123–130 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 84 (2012), 131–138. MR 2857423, DOI 10.1090/S0094-9000-2012-00856-9
- Gennady Samorodnitsky and Murad S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994. Stochastic models with infinite variance. MR 1280932
- Michel Talagrand, Les mesures vectorielles à valeurs dans $L^{0}$ sont bornées, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 4, 445–452 (1982) (French). MR 654206
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, Mathematics and its Applications (Soviet Series), vol. 14, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian and with a preface by Wojbor A. Woyczynski. MR 1435288, DOI 10.1007/978-94-009-3873-1
- 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 10.1007/BFb0074920
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
References
- 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. MR 3136844
- 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. MR 1196373
- G. Kallianpur and J. Xiong, Stochastic differential equations in infinite-dimensional spaces, Lecture notes - Monograph series, vol. 26, Institute of Mathematical Statistics, Hayward, 1995. MR 1465436
- C. Koch, Biophysics of computation: Information processing in single neurons, Computational Neuroscience, Oxford University Press, New York, 1999.
- S. Kwapień and W. A. Woyczyński, Random series and stochastic integrals: Single and multiple, Birkhäuser, Boston, 1992.
- 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.
- V. Radchenko, Integrals with respect to general stochastic measures, Institute of Mathematics, Kyiv, 1999, in Russian.
- —, On the product of a random and a real measure, Theory Probab. Math. Statist. 70 (2005), 197–206.
- —, Evolution equations driven by general stochastic measures in Hilbert space, Theory Probab. Appl. 59 (2015), 328–339. MR 3416054
- —, Averaging principle for equation driven by a stochastic measure, Stochastics 91 (2019), no. 6, 905–915. MR 3985803
- 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. MR 3914723
- —, Fourier series and Fourier – Haar series for stochastic measures, Theor. Probability and Math. Statist. (2018), no. 96, 159–167.
- V. M. Radchenko, Cable equation with a general stochastic measure, Theor. Probability and Math. Statist. (2012), no. 84, 131–138. MR 2857423
- G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian random processes, Chapman and Hall, London, 1994. MR 1280932
- M. Talagrand, Les mesures vectorielles à valeurs dans ${L}_0$ sont bornées, Annales scientifiques de l’École Normale Supérieure 14 (1981), no. 4, 445–452. MR 654206
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanian, Probability distributions on Banach spaces, D. Reidel Publishing Co., Dordrecht, 1987. MR 1435288
- J. B. Walsh, An introduction to stochastic partial differential equations, Lect. Notes Math. 1180 (1986), 265–439. MR 876085
- A. Zygmund, Trigonometric series. 3rd ed., Cambridge Univ. Press, Cambridge, 2002. MR 0236587
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
60H15,
60G57,
60H05
Retrieve articles in all journals
with MSC (2020):
60H15,
60G57,
60H05
Additional Information
B. I. Manikin
Affiliation:
Department of Mathematical Analysis, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine
Email:
bmanikin@gmail.com
V. M. Radchenko
Affiliation:
Department of Mathematical Analysis, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine
Email:
vradchenko@univ.kiev.ua
Keywords:
Stochastic measure,
stochastic cable equation,
mild solution
Received by editor(s):
December 28, 2020
Published electronically:
September 24, 2021
Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv
