Asymptotic properties of the parabolic equation driven by stochastic measure
Volume 9, Issue 4 (2022), pp. 483–498
Pub. online: 6 September 2022
Type: Research Article
Open Access
Open Access
Received
20 April 2022
20 April 2022
Revised
20 August 2022
20 August 2022
Accepted
23 August 2022
23 August 2022
Published
6 September 2022
6 September 2022
Abstract
A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as $t\to \infty $ is studied.
References
Bodnarchuk, I.M.: Regularity of the mild solution of a parabolic equation with stochastic measure. Ukr. Math. J. 69, 1–18 (2017). MR3631616. https://doi.org/10.1007/s11253-017-1344-4
Borysenko, O.D., Borysenko, D.O.: Asymptotic behavior of a solution of the non-autonomous logistic stochastic differential equation. Theory Probab. Math. Stat. 101, 39–50 (2020). MR4060330. https://doi.org/10.1090/tpms/1110
Friedman, A.: Stochastic Differential Equations and Applications. Academic Press, New York (1975). MR0494490
Ilyin, A.M., Kalashnikov, A.S., Oleynik, O.A.: Linear second-order partial differential equations of the parabolic type. J. Math. Sci. 108(4), 435–542 (2002). https://doi.org/10.1023/A:1013156322602
Jiang, Y., Wang, S., Wang, X.: Asymptotics of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. Acta Math. Sci. 39, 731–746 (2019). MR4066502. https://doi.org/10.1007/s10473-019-0307-2
Kamont, A.: A discrete characterization of Besov spaces. Approx. Theory Appl. (N. S.) 13(2), 63–77 (1997). MR1750304
Kohatsu-Higa, A., Nualart, D.: Large time asymptotic properties of the stochastic heat equation. J. Theor. Probab. 34(3), 1455–1473 (2021). MR4289891. https://doi.org/10.1007/s10959-020-01007-y
Kwapien, S., Woyczynski, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhauser, Boston (1992). MR1167198. https://doi.org/10.1007/978-1-4612-0425-1
Manikin, B.I.: Averaging principle for the one-dimensional parabolic equation driven by stochastic measure. Mod. Stoch. Theory Appl. 9(2), 123–137 (2022). MR4420680. https://doi.org/10.15559/21-vmsta195
Memin, T., Mishura, Y., Valkeia, E.: Inequalities for the moment of Wiener integrals with respect to a fractional Brownian motion. Stat. Probab. Lett. 51, 197–206 (2001). MR1822771. https://doi.org/10.1016/S0167-7152(00)00157-7
Radchenko, V.M.: Mild solution of the heat equation with a general stochastic measure. Stud. Math. 194(3), 231–251 (2009). MR2539554. https://doi.org/10.4064/sm194-3-2
Radchenko, V.M.: Evolution equations driven by general stochastic measures in Hilbert space. Theory Probab. Appl. 59(2), 328–339 (2015). MR3416054. https://doi.org/10.1137/S0040585X97T987119
Radchenko, V.M.: Averaging principle for equation driven by a stochastic measure. Stochastics 91(6), 905–915 (2019). MR3985803. https://doi.org/10.1080/17442508.2018.1559320
Radchenko, V.M., Manikin, B.I.: Approximation of the solution to the parabolic equation driven by stochastic measure. Theory Probab. Math. Stat. 102, 145–156 (2020). MR4421340. https://doi.org/10.1090/tpms
Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994). MR1280932
