Averaging principle for a stochastic cable equation
Volume 7, Issue 4 (2020), pp. 449–467
Pub. online: 21 December 2020
Type: Research Article
Open Access
Open Access
Received
5 October 2020
5 October 2020
Revised
22 November 2020
22 November 2020
Accepted
1 December 2020
1 December 2020
Published
21 December 2020
21 December 2020
Abstract
We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.
References
Ahmed, H.M., Zhu, Q.: The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps. Appl. Math. Lett. 112 (2021). MR4151504. https://doi.org/10.1016/j.aml.2020.106755
Avetisian, D., Ralchenko, K.: Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation. Mod. Stoch. Theory Appl. 7(3), 339–356 (2020). MR4159153
Bao, J., Yin, G., Yuan, C.: Two-time-scale stochastic partial differential equations driven by α-stable noises: averaging principles. Bernoulli 23(1), 645–669 (2017). MR3556788. https://doi.org/10.3150/14-BEJ677
Bodnarchuk, I.M.: Regularity of the mild solution of a parabolic equation with random measure. Ukr. Math. J. 69(1), 1–18 (2017). MR3631616. https://doi.org/10.1007/s11253-017-1344-4
Bodnarchuk, I.M.: Wave equation with a stochastic measure. Theory Probab. Math. Stat. 94, 1–16 (2017). MR3553450. https://doi.org/10.1090/tpms/1005
Bodnarchuk, I.M., Radchenko, V.M.: Wave equation in the plane driven by a general stochastic measure. Theory Probab. Math. Stat. 98, 73–90 (2019). MR3992992
Bodnarchuk, I.M., Radchenko, V.M.: The wave equation in the three-dimensional space driven by a general stochastic measure. Theory Probab. Math. Stat. 100, 43–60 (2020). MR3992992
Bodnarchuk, I.M., Shevchenko, G.M.: Heat equation in a multidimensional domain with a general stochastic measure. Theory Probab. Math. Stat. 93, 1–17 (2016). MR3553436. https://doi.org/10.1090/tpms/991
Bréhier, C.-E.: Strong and weak orders in averaging for SPDEs. Stoch. Process. Appl. 122(7), 2553–2593 (2012). MR2926167. https://doi.org/10.1016/j.spa.2012.04.007
Bréhier, C.-E.: Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component. Stoch. Process. Appl. 130(6), 3325–3368 (2020). MR4092407. https://doi.org/10.1016/j.spa.2019.09.015
Cerrai, S., Lunardi, A.: Averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations: The almost periodic case. SIAM J. Math. Anal. 49(4), 2843–2884 (2017). MR3679916. https://doi.org/10.1137/16M1063307
Fu, H., Liu, J.: Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations. J. Math. Anal. Appl. 384(1), 70–86 (2011). MR2822851. https://doi.org/10.1016/j.jmaa.2011.02.076
Fu, H., Wan, L., Liu, J., Liu, X.: Weak order in averaging principle for stochastic wave equation with a fast oscillation. Stoch. Process. Appl. 128, 2557–2580 (2018). MR3811697. https://doi.org/10.1016/j.spa.2017.09.021
Gao, P.: Averaging principle for the higher order nonlinear Schrödinger equation with a random fast oscillation. J. Stat. Phys. 171, 897–926 (2018). MR3800899. https://doi.org/10.1007/s10955-018-2048-3
Kamont, A.: A discrete characterization of Besov spaces. Approx. Theory Appl. 13(2), 63–77 (1997). MR1750304
Kwapień, S., Woyczyński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992). MR1167198. https://doi.org/10.1007/978-1-4612-0425-1
Liu, W., Röckner, M., Sun, X., Xie, Y.: Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients. J. Differ. Equ. 268(6), 2910–2948 (2020). MR4047972. https://doi.org/10.1016/j.jde.2019.09.047
Pryhara, L., Shevchenko, G.: Stochastic wave equation in a plane driven by spatial stable noise. Mod. Stoch. Theory Appl. 3(3), 237–248 (2016). MR3576308. https://doi.org/10.15559/16-VMSTA62
Pryhara, L.I., Shevchenko, G.M.: Wave equation with a stable noise. Theory Probab. Math. Stat. 96, 145–157 (2018). MR3666878. https://doi.org/10.1090/tpms/1040
Radchenko, V.: 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.: 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
Radchenko, V.: Strong convergence rate in averaging principle for the heat equation driven by a general stochastic measure. Commun. Stoch. Anal. 13(2) (2019). MR4002769. https://doi.org/10.31390/cosa.13.2.01
Radchenko, V., Stefans’ka, N.: Approximation of solutions of the stochastic wave equation by using the Fourier series. Mod. Stoch. Theory Appl. 5(4), 429–444 (2018). MR3914723. https://doi.org/10.15559/18-vmsta115
Radchenko, V.M.: Cable equation with a general stochastic measure. Theory Probab. Math. Stat. 84, 131–138 (2012). MR2857423. https://doi.org/10.1090/S0094-9000-2012-00856-9
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.N.: Evolution equations with general stochastic measures in Hilbert space. Theory Probab. Appl. 59(2), 328–339 (2015). MR3416054. https://doi.org/10.1137/S0040585X97T987119
Ralchenko, K., Shevchenko, G.: Existence and uniqueness of mild solution to fractional stochastic heat equation. Mod. Stoch. Theory Appl. 6(1), 57–79 (2019). MR3935427. https://doi.org/10.15559/18-vmsta122
Shen, G., Wu, J.-L., Yin, X.: Averaging principle for fractional heat equations driven by stochastic measures. Appl. Math. Lett. 106 (2020). MR4090373. https://doi.org/10.1016/j.aml.2020.106404
Shen, G., Xiao, R., Yin, X.: Averaging principle and stability of hybrid stochastic fractional differential equations driven by Lévy noise. Int. J. Syst. Sci. 51(12), 2115–2133 (2020). MR4141427. https://doi.org/10.1080/00207721.2020.1784493
Tuckwell, H.C.: Introduction to Theoretical Neurobiology. Volume 1. Linear Cable Theory and Dendritic Structure. Cambridge University Press, New York (1988). MR0947345
Walsh, J.B.: An introduction to stochastic partial differential equations. In: Lecture Notes in Mathematics vol. 1180, pp. 265–439. Springer, Berlin (1986). MR0876085. https://doi.org/10.1007/BFb0074920
Wang, P., Xu, Y.: Averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion. J. Funct. Spaces 2020 (2020). MR4108143. https://doi.org/10.1155/2020/5212690
