非局部随机扩散方程解的HO¨lder连续性
HO¨lder Continuous of the Solutions to Nonlocal Stochastic Diffusion Equations
DOI: 10.12677/PM.2023.1312351, PDF,    科研立项经费支持
作者: 贾 倩*, 王 伟：天津师范大学数学科学学院，天津
关键词: 分数布朗运动；HO¨lder连续性；L∞估计；尾估计；Fractional Brownian Motion； HO¨lder Continuity； L∞ Estimates； Tail Estimates
摘要: 本文的目标是获得非局部随机扩散方程解的Hölder连续性。利用Campanato估计和Sobolev嵌入定理，首先证明了非局部随机扩散方程的温和解的Hölder连续性，即解u属于空间Cβ(DT;Lp(Ω))。其次，通过使用尾估计，得到了Lp(Ω;Cβ*(DT)中的温和解的估计。
Abstract: In this paper, we aim to obtain the Hölder continuous of solutions to nonlocal stochastic equations. By using Campanato estimates and Sobolev embedding theorem, we first prove the Hölder con-tinuous of the mild solution of nonlocalstochastic diffusion equations in the sense that the solution u belongs to the space Cβ(DT;Lp(Ω)). Then by using tail estimates, we obtain the estimates of the mild solution in Lp(Ω;Cβ*(DT).
文章引用：贾倩, 王伟. 非局部随机扩散方程解的HO¨lder连续性[J]. 理论数学, 2023, 13(12): 3380-3394. https://doi.org/10.12677/PM.2023.1312351

参考文献

[1] Kim, I. (2015) A BMO Estimate for Stochastic Singular Integral Operators and Its Application to SPDEs. Journal of Functional Analysis, 269, 1289-1309. [Google Scholar] [CrossRef] 
[2] Lv, G., Gao, H., Wei, J. and Wu, J.L. (2019) BMO and Morrey-Campanato Estimates for Stochastic Convolutions and Schauder Estimates for Sto-chastic Parabolic Equations. Differential Equations, 266, 2666-2717. [Google Scholar] [CrossRef] 
[3] Krylov, N.V. (1999) An Analytic Approach to SPDEs. In: Sto-chastic Partial Differential Equations: Six Perspectives, Vol. 64, AMS, Providence, 185-242. [Google Scholar] [CrossRef] 
[4] Krylov, N.V. (1996) On Lp-Theory of Stochastic Partial Differential Equations in the Whole Space. SIAM Journal on Mathematical Analysis, 27, 313-340. [Google Scholar] [CrossRef] 
[5] Kim, I. and Kim, K.H. (2016) An Lp-Theory for Stochastic Partial Differential Equations Driven by Lévy Processes with Pseudo-Differential Operators of Arbitrary Order. Sto-chastic Processes and their Applications, 126, 2761-2786. [Google Scholar] [CrossRef] 
[6] Zhang, X. (2006) Lp-Theory of Semi-Linear SPDEs on General Measure Spaces and Applications. Journal of Functional Analysis, 239, 44-75. [Google Scholar] [CrossRef] 
[7] Zhang, X. (2013) Lp-Maximal Regularity of Nonlocal Parabolic Equations and Applications. Annales de l’Institut Henri Poincaré C, 30, 573-614. [Google Scholar] [CrossRef] 
[8] Duncan, T.E., Pasik-Duncan, B. and Maslowski, B. (2002) Fractional Brownian Motion and Stochastic Equations in Hilbert Spaces. Stochastics and Dynamics, 2, 225-250. [Google Scholar] [CrossRef] 
[9] Comte, F. and Renault, E. (1996) Long Memory Continuous Time Models. Journal of Econometrics, 73, 101-149. [Google Scholar] [CrossRef] 
[10] Boufoussi, B. and Hajji, S. (2012) Neutral Stochastic Func-tional Differential Equations Driven by a Fractional Brownian Motion in a Hilbert Space. Statistics and Probability Letters, 82, 1549-1558. [Google Scholar] [CrossRef] 
[11] Caraballo, T., Garrido-Atienza, M.J. and Taniguchi, T. (2011) The Existence and Exponential Behavior of Solutions to Stochastic Delay Evolution Equations with a Fractional Brownian Motion. Nonlinear Analysis: Theory, Methods and Applications, 74, 3671-3684. [Google Scholar] [CrossRef] 
[12] Li, Z. and Yan, L. (2019) Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion. Journal of Theoretical Probability, 32, 1399-1419. [Google Scholar] [CrossRef] 
[13] Xu, L., Li, Z. and Luo, J. (2017) Global Attracting Set and Ex-ponential Decay of Second-Order Neutral Stochastic Functional Differential Equations Driven by fBm. Advances in Difference Equations, 2017, Article No. 134. [Google Scholar] [CrossRef] 
[14] Hsu, E.P., Wang, Y. and Wang, Z. (2017) Stochastic De Giorgi Iteration and Regularity of Stochastic Partial Differential Equations. The Annals of Probability, 45, 2855-2866. [Google Scholar] [CrossRef] 
[15] Du, K. and Liu, J. (2016) A Schauder Estimate for Stochastic PDEs. Comptes Rendus Mathematique, 354, 371-375. [Google Scholar] [CrossRef] 
[16] Debussche, A., De Moor, S. and Hofmanová, M. (2015) A Reg-ularity Result for Quasilinear Stochastic Partial Differential Equations of Parabolic Type. SIAM Journal on Mathematical Analysis, 47, 1590-1614. [Google Scholar] [CrossRef] 
[17] Kuksin, S.B., Nadirashvili, N.S. and Piatnitski, A.L. (2003) Hölder Esti-mates for Solutions of Parabolic SPDEs. Theory of Probability and Its Applications, 47, 157-163. [Google Scholar] [CrossRef] 
[18] Tian, R., Ding, L., Wei, J. and Zheng, S. (2019) Hölder Esti-mates of Mild Solutions for Nonlocal SPDEs. Advances in Difference Equations, 2019, Article No. 159. [Google Scholar] [CrossRef] 
[19] Lv, G., Gao, H., Wei, J. and Wu, J.L. (2023) On the Campanato and Hölder Regularity of Local and Nonlocal Stochastic Diffusion Equations. Discrete and Continuous Dynamical Systems-B, 28, 1244. [Google Scholar] [CrossRef] 
[20] Chen, Y.Z. (2003) Second Order Parabolic Partial Differential Equa-tions. Peking University Press, Beijing.
[21] Bogdan, K. and Jakubowski, T. (2007) Estimates of Heat Kernel of Frac-tional Laplacian Perturbed by Gradient Operators. Communications in Mathematical Physics, 271, 179-198. [Google Scholar] [CrossRef] 
[22] Bogdan, K., Stós, A. and Sztonyk, P. (2003) Harnack Inequality for Stable Processes on d-Sets. Studia Mathematica, 158, 163-198. [Google Scholar] [CrossRef] 
[23] Chen, Z.Q. and Hu, E. (2015) Heat Kernel Estimates for Δ+Δα/2 under Gradient Perturbation. Stochastic Processes and Their Applications, 125, 2603-2642. [Google Scholar] [CrossRef] 
[24] Imbert, C. (2005) A Non-Local Regularization of First Order Hamilton-Jacobi Equations. Journal of Differential Equations, 211, 218-246. [Google Scholar] [CrossRef] 
[25] Biagini, F., Hu, Y., Φksendal, B. and Zhang, T. (2008) Stochastic Calculus for Fractional Brownian Motion and Applications. Springer Science and Business Media, Berlin. [Google Scholar] [CrossRef] 

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