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

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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
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- 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.
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*Approximation of the solution to the parabolic equation driven by stochastic measure*, Theory Probab. Math. Statist. **102** (2020), in press.
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*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**
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*Fourier series and Fourier – Haar series for stochastic measures*, Theor. Probability and Math. Statist. (2018), no. 96, 159–167.
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*Stable non-Gaussian random processes*, Chapman and Hall, London, 1994. MR **1280932**
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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