The Burgers-type equation driven by a stochastic measure

Author:
Vadym Radchenko

Journal:
Theor. Probability and Math. Statist. **110** (2024), 185-199

MSC (2020):
Primary 60H15; Secondary 60G57

DOI:
https://doi.org/10.1090/tpms/1213

Published electronically:
May 10, 2024

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Additional Information

Abstract: We study the one-dimensional equation driven by a stochastic measure $\mu$. For $\mu$ we assume only $\sigma$-additivity in probability. Our results imply the global existence and uniqueness of the solution to the heat equation and the local existence and uniqueness of the solution to the Burgers equation. The averaging principle for such equation is studied.

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References
- I. Bodnarchuk,
*Regularity of the mild solution of a parabolic equation with stochastic measure*, Ukr. Math. J. **69** (2017), 1–18. MR **3631616**
- —,
*Averaging principle for a stochastic cable equation*, Mod. Stoch. Theory Appl. **7** (2020), no. 4, 449–467. MR **4195646**
- I. Bodnarchuk and V. Radchenko,
*The wave equation in the three-dimensional space driven by a general stochastic measure*, Theor. Probability and Math. Statist. **100** (2020), 43–60. MR **3992992**
- Zh. Dong and T.G. Xu,
*One-dimensional stochastic Burgers equation driven by Lévy processes*, J. Funct. Anal. **243** (2007), 631–678. MR **2289699**
- I. Gyöngy,
*Existence and uniqueness results for semilinear stochastic partial differential equations*, Stochastic Process. Appl. **73** (1998), 271–299. MR **1608641**
- I. Gyöngy and D. Nualart,
*On the stochastic Burgers equation in the real line*, Ann. Probab. **27** (1999), 782–802. MR **1698967**
- I. Gyöngy and C. Rovira,
*On stochastic partial differential equations with polynomial nonlinearities*, Stochastics **67** (1999), 123–146. MR **1717799**
- N. Jacob, A. Potrykus, and J.-L. Wu,
*Solving a non-linear stochastic pseudo-differential equation of Burgers type*, Stochastic Proc. Appl. **120** (2010), 2447–2467. MR **2728173**
- S. Kwapień and W. A. Woyczyński,
*Random series and stochastic integrals: Single and multiple*, Birkhäuser, Boston, 1992. MR **1167198**
- P. Lewis and D. Nualart,
*Stochastic Burgers’ equation on the real line: regularity and moment estimates*, Stochastics **90** (2018), 1053–1086. MR **3854527**
- B. Manikin,
*Averaging principle for the one-dimensional parabolic equation driven by stochastic measure*, Mod. Stoch. Theory Appl. **9** (2022), no. 2, 123–137. MR **4420680**
- S. Mazzonetto and D. Salimova,
*Existence, uniqueness, and numerical approximations for stochastic Burgers equations*, Stochastic Anal. Appl. **38** (2020), 623–646. MR **4112739**
- 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), 197–206. MR **1822771**
- S. Peszat and J. Zabczyk,
*Stochastic partial differential equations with Lévy noise: An evolution equation approach*, Cambridge University Press, Cambridge, 2007. MR **2356959**
- V. Radchenko,
*Mild solution of the heat equation with a general stochastic measure*, Studia Math. **194** (2009), 231–251. MR **2539554**
- —,
*Evolution equations driven by general stochastic measures in Hilbert space*, Theory Probab. Appl. **59** (2015), 328–339. MR **3416054**
- —,
*Averaging principle for the heat equation driven by a general stochastic measure*, Statist. Probab. Lett. **146** (2019), 224–230. MR **3885229**
- —,
*Strong convergence rate in averaging principle for the heat equation driven by a general stochastic measure*, Commun. Stoch. Anal. **13** (2019), 1–17. MR **4002769**
- —,
*General stochastic measures: Integration, path properties, and equations*, Wiley–ISTE, London, 2022.
- —,
*The Burgers equation driven by a stochastic measure*, Mod. Stoch. Theory Appl. (2023), 1–18.
- —,
*Transport equation driven by a stochastic measure*, Mod. Stoch. Theory Appl. (2023), 1–13.
- G. Samorodnitsky and M. S. Taqqu,
*Stable non-Gaussian random processes*, Chapman and Hall, London, 1994. MR **1280932**
- G. Shen, J.-L. Wu, and X. Yin,
*Averaging principle for fractional heat equations driven by stochastic measures*, Appl. Math. Lett. **106** (2020), 106404. MR **4090373**
- C. Tudor,
*On the Wiener integral with respect to a sub-fractional Brownian motion on an interval*, J. Math. Anal. Appl. **351** (2009), 456–468. MR **2472957**
- Sh. Yuan, D. Blömker, and J. Duan,
*Stochastic turbulence for Burgers equation driven by cylindrical Lévy process*, Stoch. Dyn. **22** (2022), 2240004. MR **4431443**
- G. Zhou, L. Wang, and J.-L. Wu,
*Global well-posedness of 2D stochastic Burgers equations with multiplicative noise*, Statist. Probab. Lett. **182** (2022), 109315. MR **4347488**

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Additional Information

**Vadym Radchenko**

Affiliation:
Department of Mathematical Analysis, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, Kyiv, Ukraine, 01601

Email:
vadymradchenko@knu.ua

Keywords:
Stochastic Burgers equation,
stochastic heat equation,
stochastic measure,
mild solution,
averaging principle

Received by editor(s):
November 8, 2022

Accepted for publication:
March 26, 2023

Published electronically:
May 10, 2024

Additional Notes:
This work was supported by Alexander von Humboldt Foundation, grant 1074615.

Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv