Approximation of the solution to the parabolic equation driven by stochastic measure
Authors:
B. I. Manikin and V. M. Radchenko
Journal:
Theor. Probability and Math. Statist. 102 (2020), 145-156
MSC (2020):
Primary 60H15; Secondary 60G57, 60H05
DOI:
https://doi.org/10.1090/tpms/1131
Published electronically:
March 29, 2021
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Additional Information
Abstract: The one-dimensional parabolic equation driven by general stochastic measure $\mu$ is considered. For $\mu$ we assume only $\sigma$-additivity in probability, coefficients of the parabolic operator of the equation do not depend on space variable. It is proved that the convergence of stochastic integrators implies the convergence of the respective solutions.
References
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References
- I. M. Bodnarchuk, Regularity of the mild solution of a parabolic equation with stochastic measure, Ukr. Math. J. 69 (2017), no. 1, 1–18. MR 3631616
- V. M. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554
- O. O. Vertsimakha and V. M. Radchenko, Mild solution of the parabolic equation driven by a $\sigma$-finite stochastic measure, Theor. Probab. Math. Statist. 97 (2018), 17–32. MR 3745996
- Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stoch. Proc. Appl. 55 (1995), no. 2, 329–358. MR 1313027
- L. Pryhara and G. Shevchenko, Approximations for a solution to stochastic heat equation with stable noise, Mod. Stoch.: Theory Appl. 3 (2016), no. 2, 133–144. MR 3519720
- V. M. Radchenko, Fourier series expansion of stochastic measures, Theory Probab. Appl. 63 (2018), no. 2, 318–326. MR 3796494
- V. M. Radchenko and N. O. Stefans’ka, Approximation of solutions of wave equation driven by stochastic measure, Theor. Probab. Math. Statist. 99 (2018), 203–211. (Ukrainian)
- V. Radchenko and N. 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
- F. J. Delgado-Vences and M. Sanz-Solé, Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm, Bernoulli 20 (2014), no. 4, 2169–2216. MR 3263102
- F. J. Delgado-Vences and M. Sanz-Solé, Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22 (2016), no. 3, 1572–1597. MR 3474826
- E. G. F. Thomas. Vector integration, Quaestiones Mathematicae, 35 (2012), no. 4, 391–416. MR 2999997
- S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992.
- V. N. Radchenko, Integrals with respect to general stochastic measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
- M. Talagrand, Les mesures vectorielles a valeurs dans $L_0$ sont bornées, Ann. Sci. École Norm. Sup., 14 (1981), no. 4, 445–452. MR 654206
- T. Memin, Yu. Mishura, and E. 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
- G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, Chapman and Hall, London, 1994. MR 1280932
- V. Radchenko, Averaging principle for equation driven by a stochastic measure, Stochastics 91 (2019), no. 6, 905–915. MR 3985803
- V. M. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Theor. Prob. Appl. 59 (2015), no. 2, 328–339. MR 3416054
- A. M. Ilyin, A. S. Kalashnikov, and O. A. Oleynik, Linear second-order partial differential equations of the parabolic type, J. Math. Sci. 108 (2002), no. 4, 435–542.
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanian, Probability distributions on Banach spaces, D. Reidel Publishing Co., Dordrecht, 1987. MR 1435288
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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 parabolic equation,
mild solution.
Received by editor(s):
November 29, 2019
Published electronically:
March 29, 2021
Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv
