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Averaging principle for the one-dimensional parabolic equation driven by stochastic measure
Volume 9, Issue 2 (2022), pp. 123–137
Boris Manikin ORCID icon link to view author Boris Manikin details  

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https://doi.org/10.15559/21-VMSTA195
Pub. online: 10 January 2022      Type: Research Article      Open accessOpen Access

Received
2 June 2021
Revised
27 November 2021
Accepted
27 November 2021
Published
10 January 2022

Abstract

A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure is considered. The averaging principle for the equation is established. The convergence rate is compared with other results on related topics.

References

[1] 
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
[2] 
Bodnarchuk, I.M.: Averaging principle for a stochastic cable equation. Mod. Stoch. Theory Appl. 7, 449–467 (2020). MR4195646. https://doi.org/10.15559/20-vmsta168
[3] 
Bodnarchuk, I.M., Radchenko, V.M.: The equation for vibrations of a fixed string driven by a general stochastic measure. Theory Probab. Math. Stat. 101, 1–11 (2020). MR3824679. https://doi.org/10.1090/tpms/1108
[4] 
Brehier, C.-E.: Strong and weak orders in averaging for SPDEs. Stoch. Process. Appl. 122, 2553–2593 (2012). MR2926167. https://doi.org/10.1016/j.spa.2012.04.007
[5] 
Bressloff, P.C.: Waves in Neutral Media: From Single Neurons to Neural Fields. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York (2014). MR3136844. https://doi.org/10.1007/978-1-4614-8866-8
[6] 
Duan, J., Wang, W.: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, Amsterdam (2014). MR3289240
[7] 
Fu, H., Liu, J.: Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations. J. Math. Anal. Appl. 384, 70–86 (2011). MR2822851. https://doi.org/10.1016/j.jmaa.2011.02.076
[8] 
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). MR1875963. https://doi.org/10.1023/A:1013156322602
[9] 
Kamont, A.: A discrete characterization of Besov spaces. Approx. Theory Appl. (N.S.) 13(2), 63–77 (1997). MR1750304
[10] 
Kwapien, S., Woyczynski, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992). MR1167198. https://doi.org/10.1007/978-1-4612-0425-1
[11] 
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
[12] 
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
[13] 
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
[14] 
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
[15] 
Radchenko, V.M.: Averaging principle for the heat equation driven by a general stochastic measure. Stat. Probab. Lett. 146, 224–230 (2019). MR3885229. https://doi.org/10.1016/j.spl.2018.11.024
[16] 
Radchenko, V.M.: Strong convergence rate in averaging principle for the heat equation driven by a general stochastic measure. Commun. Stoch. Anal. 13, 1 (2019). https://doi.org/10.31390/cosa.13.2.01
[17] 
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). https://doi.org/10.1090/tpms/1131
[18] 
Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994). MR1280932
[19] 
Shen, G., Wu, J.-L., Yin, X.: Averaging principle for fractional heat equations driven by stochastic measures. Appl. Math. Lett. 106, 106404 (2020). MR4090373. https://doi.org/10.1016/j.aml.2020.106404
[20] 
Wang, W., Roberts, A.: Average and deviation for slow-fast stochastic partial differential equations. J. Differ. Equ. 253, 1265–1286 (2012). MR2927381. https://doi.org/10.1016/j.jde.2012.05.011

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Keywords
Stochastic measure averaging principle mild solution stochastic parabolic equation

MSC2010
60G57

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