Theory of Probability and Mathematical Statistics
The Burgers-type equation on a line segment driven by a stochastic measure
Vadym Radchenko
Link
Abstract: We study the class of one-dimensional equations on a line segment driven by a stochastic measure μ. For μ, we assume only σ-additivity in probability. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are studied, and, for some cases, the Hölder continuity of the solution is established. The equivalence of weak and mild forms of the equation is proved.
Keywords: Stochastic measure, stochastic Burgers equation, stochastic heat equation, mild solution, weak solution
Bibliography: Ī. M. Bodnarchuk, Regularity of the mild solution of a parabolic equation with a random measure, Ukraïn. Mat. Zh. 69 (2017), no. 1, 3–16 (Ukrainian, with English and Russian summaries); English transl., Ukrainian Math. J. 69 (2017), no. 1, 1–18. MR 3631616, DOI 10.1007/s11253-017-1344-4
D. J.-M. Candil, Localization errors of the stochastic heat equation, EPFL Ph. D. Thesis (2022), 221.
Z. Dong and T. G. Xu, One-dimensional stochastic Burgers equation driven by Lévy processes, J. Funct. Anal. 243 (2007), no. 2, 631–678. MR 2289699, DOI 10.1016/j.jfa.2006.09.010
István Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Process. Appl. 73 (1998), no. 2, 271–299. MR 1608641, DOI 10.1016/S0304-4149(97)00103-8
István Gyöngy and Carles Rovira, On stochastic partial differential equations with polynomial nonlinearities, Stochastics Stochastics Rep. 67 (1999), no. 1-2, 123–146. MR 1717799, DOI 10.1080/17442509908834205
Martin Hairer and Jochen Voss, Approximations to the stochastic Burgers equation, J. Nonlinear Sci. 21 (2011), no. 6, 897–920. MR 2860933, DOI 10.1007/s00332-011-9104-3
Stanisław Kwapień and Wojbor A. Woyczyński, Random series and stochastic integrals: single and multiple, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1167198, DOI 10.1007/978-1-4612-0425-1
Sara Mazzonetto and Diyora Salimova, Existence, uniqueness, and numerical approximations for stochastic Burgers equations, Stoch. Anal. Appl. 38 (2020), no. 4, 623–646. MR 4112739, DOI 10.1080/07362994.2019.1709503
Jean Mémin, Yulia Mishura, and Esko 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, DOI 10.1016/S0167-7152(00)00157-7
S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, Encyclopedia of Mathematics and its Applications, vol. 113, Cambridge University Press, Cambridge, 2007. An evolution equation approach. MR 2356959, DOI 10.1017/CBO9780511721373
Mark A. Pinsky, Partial differential equations and boundary-value problems with applications, Pure and Applied Undergraduate Texts, vol. 15, American Mathematical Society, Providence, RI, 2011. Reprint of the third (1998) edition. MR 2849590
Vadym Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554, DOI 10.4064/sm194-3-2
V. Radchenko, Riemann integral of a random function and the parabolic equation with a general stochastic measure, Theory Probab. Math. Statist. 87 (2013), 185–198. Translation of Teor. Ǐmovīr. Mat. Stat. No. 87 (2012), 163–175. MR 3241455, DOI 10.1090/S0094-9000-2014-00912-6
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
Vadym M. Radchenko, General stochastic measures—integration, path properties and equations, Mathematics and Statistics Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2022. MR 4687103, DOI 10.1002/9781394163939
Vadym Radchenko, The Burgers equation driven by a stochastic measure, Mod. Stoch. Theory Appl. 10 (2023), no. 3, 229–246. MR 4608186, DOI 10.15559/23-vmsta224
Vadym Radchenko, The Burgers-type equation driven by a stochastic measure, Theory Probab. Math. Statist. 110 (2024), 185–199. MR 4751817, DOI 10.1090/tpms/1213
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
Guoli Zhou, Lidan Wang, and Jiang-Lun Wu, Global well-posedness of 2D stochastic Burgers equations with multiplicative noise, Statist. Probab. Lett. 182 (2022), Paper No. 109315, 6. MR 4347488, DOI 10.1016/j.spl.2021.109315
