# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022119
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## On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

 1 College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China 2 School of Mathematics and Computational Sciences, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China 3 School of Mathematics, School of Mathematical Science, Southeast University, Nanjing 211189, China 4 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China 5 Department of Mathematics, Computational Foundry, Swansea University, Swansea SA1 8EN, UK

* Corresponding author: Guangying Lv

Received  January 2022 Revised  April 2022 Early access June 2022

Fund Project: This work is partially supported by China NSF Grant Nos. 12171247, 12171084, 11771123, 11501577 and the Startup Foundation for Introducing Talent of NUIST

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Campanato estimates and Sobolev embedding theorem, we first show the Hölder continuity (locally in the whole state space $\mathbb{R}^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions belong to the space $C^{\gamma}(D_T;L^p(\Omega))$ with the optimal Hölder continuity index $\gamma$ (which is given explicitly), where $D_T: = [0, T]\times D$ for $T>0$, and $D\subset\mathbb{R}^d$ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $L^p(\Omega;C^{\gamma^*}(D_T))$. What's more, we give an explicit formula between the two indexes $\gamma$ and $\gamma^*$. Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criterion to justify Hölder continuity for the solutions on bounded domains. The novelty of this paper is that our method is suitable to the case of space-time white noise.

Citation: Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022119
##### References:
 [1] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179–198. doi: 10.1007/s00220-006-0178-y. [2] K. Bogdan, A. Stós and P. Sztonyk, Harnack inequality for stable processes on $d$-sets, Studia Math., 158 (2003), 163–198. doi: 10.4064/sm158-2-5. [3] Y. Z. Chen, Second Order Parabolic Partial Differential Equations, Peking University Press, 2003. [4] Z.-Q. Chen and E. Hu, Heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ under gradient perturbation, Stochastic Process. Appl., 125 (2015), 2603-2642.  doi: 10.1016/j.spa.2015.02.016. [5] P. A. Cioica-Licht, K.-H. Kim and K. Lee, On the regularity of the stochastic heat equation on polygonal domains in $R^2$, J. Differential Equations, 267 (2019), 6447–6479. doi: 10.1016/j.jde.2019.06.027. [6] A. Debussche, S. de Moor and M. Hofmanov$\acute{a}$, A regularity result for quasilinear stochastic partial differential equations of parabolic type, SIAM J. Math. Anal., 47 (2015), 1590–1614. doi: 10.1137/130950549. [7] L. Denis, A. Matoussi and L. Stoica, $L^p$ estimates for the uniform norm of solutions of quasilinear SPDE's, Probab. Theory Related Fields, 133 (2005), 437–463. doi: 10.1007/s00440-005-0436-5. [8] K. Du and J. Liu, On the Cauchy problem for stochastic parabolic equations in Hölder spaces, Trans. Amer. Math. Soc., 371 (2019), 2643–2664. doi: 10.1090/tran/7533. [9] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493–540. doi: 10.1142/S0218202512500546. [10] E. P. Hsu, Y. Wang and Z. Wang, Stochastic De Giorgi iteration and regularity of stochastic partial differential equations, Ann. Probab., 45 (2017), 2855–2866. doi: 10.1214/16-AOP1126. [11] C. Imbert, A non-local regularization of first order Hamilton-Jacobi equations, J. Differential Equations, 211 (2005), 218–246. doi: 10.1016/j.jde.2004.06.001. [12] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferential functions further results, Comput. Math. Appl., 51 (2006), 1367–1376. doi: 10.1016/j.camwa.2006.02.001. [13] I. Kim, A BMO estimate for stochastic singular integral operators and its application to SPDEs, J. Funct. Anal., 269 (2015), 1289-1309.  doi: 10.1016/j.jfa.2015.05.015. [14] I. Kim and K.-H. Kim, An $L_p$-theory for stochastic partial differential equations driven by Lévy processes with pseudo-differential operators of arbitrary order, Stochastic Process. Appl., 126 (2016), 2761-2786.  doi: 10.1016/j.spa.2016.03.001. [15] I. Kim, K.-H. Kim and S. Lim, Parabolic Littlewood-Paley inequality for a class of time-dependent pseudo-differential operators of arbitrary order, and applications to high-order stochastic PDE, J. Math. Anal. Appl., 436 (2016), 1023-1047.  doi: 10.1016/j.jmaa.2015.12.040. [16] K.-H. Kim, $L_q (L_p)$ theory and Hölder estimates for parabolic SPDEs, Stochastic Process. Appl., 114 (2004), 313–330. doi: 10.1016/j.spa.2004.07.004. [17] K.-H. Kim and P. Kim, An $L_p$-theory of a class of stochastic equations with the random fractional Laplacian driven by Lévy processes, Stochastic Process. Appl., 122 (2012), 3921-3952.  doi: 10.1016/j.spa.2012.08.001. [18] K.-H. Kim, K. Lee and J. Seo, A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic in $\mathbb{R}^d$, J. Differential Equations, 291 (2021), 154-194.  doi: 10.1016/j.jde.2021.05.001. [19] K.-H. Kim, D. Park and J. Ryu, An $L_q(L_p)$-theory for diffusion equations with space-time nonlocal operators, J. Differential Equations, 287 (2021), 376-427.  doi: 10.1016/j.jde.2021.04.003. [20] N. V. Krylov, An analytic approach to SPDEs, in: Stochastic Partial Differential Equations: Six Perspectives, in: Math. Surveys Monogr., 64 (1999), 185-242. doi: 10.1090/surv/064/05. [21] N. V. Krylov, On $L_p$-theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal., 27 (1996), 313–340. doi: 10.1137/S0036141094263317. [22] S. B. Kuksin, N. S. Nadirashvili and A. L. Piatnitski, Hölder estimates for solutions of parabolic SPDEs, Theory Probab. Appl., 47 (2003), 157–164. doi: 10.1137/S0040585X97979524. [23] K. Li and J. Peng, Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941–959. doi: 10.1007/s00033-013-0369-2. [24] G. Lv, H. Gao, J. Wei and J.-L. Wu, BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666–2717. doi: 10.1016/j.jde.2018.08.042. [25] G. Lv, H. Gao, J. Wei and J.-L. Wu, Hölder estimates for solutions of stochastic nonlocal diffusion equations, Stochastic PDEs and Modelling of Multiscale Complex System, 97–110, Interdiscip. Math. Sci., 20, World Sci. Publ., Hackensack, NJ, 2019. [26] J. Neerven, M. C. Veraar and L. Weis, Maximal $L_p$-Regularity for Stochastic Evolution Equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. [27] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag Berlin, 1991. doi: 10.1007/978-3-662-21726-9. [28] R. Tian, L. Ding, J. Wei and S. Zheng, Hölder estimates of mild solutions for nonlocal SPDEs, Adv. Difference Equ., (2019), Paper No. 159, 12 pp. doi: 10.1186/s13662-019-2097-1. [29] J. van Neerven, M. Veraar and L. Weis, Stochastic maximal $L_p$regularity, Ann. Probab., 40 (2012), 788-812.  doi: 10.1214/10-AOP626. [30] X. Wang, Hölder continuous of the solutions to stochastic nonlocal heat equations, Comput. Math. Appl., 78 (2019), 741–753. doi: 10.1016/j.camwa.2019.02.036. [31] J. Wei, G. Lv and W. Wang, Stochastic transport equation with bounded and Dini continuous drift, J. Differential Equations, 323 (2022), 359–403. doi: 10.1016/j.jde.2022.03.038. [32] X. Xie, J. Duan, X. Li and G. Lv, A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift, arXiv: 1504.04631. [33] X. Zhang, $L_p$-theory of semi-linear SPDEs on general measure spaces and applications, J. Funct. Anal., 239 (2006), 44-75.  doi: 10.1016/j.jfa.2006.01.014. [34] X. Zhang, $L_p$-maximal regularity of nonlocal parabolic equations and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 573-614.  doi: 10.1016/j.anihpc.2012.10.006.

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##### References:
 [1] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179–198. doi: 10.1007/s00220-006-0178-y. [2] K. Bogdan, A. Stós and P. Sztonyk, Harnack inequality for stable processes on $d$-sets, Studia Math., 158 (2003), 163–198. doi: 10.4064/sm158-2-5. [3] Y. Z. Chen, Second Order Parabolic Partial Differential Equations, Peking University Press, 2003. [4] Z.-Q. Chen and E. Hu, Heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ under gradient perturbation, Stochastic Process. Appl., 125 (2015), 2603-2642.  doi: 10.1016/j.spa.2015.02.016. [5] P. A. Cioica-Licht, K.-H. Kim and K. Lee, On the regularity of the stochastic heat equation on polygonal domains in $R^2$, J. Differential Equations, 267 (2019), 6447–6479. doi: 10.1016/j.jde.2019.06.027. [6] A. Debussche, S. de Moor and M. Hofmanov$\acute{a}$, A regularity result for quasilinear stochastic partial differential equations of parabolic type, SIAM J. Math. Anal., 47 (2015), 1590–1614. doi: 10.1137/130950549. [7] L. Denis, A. Matoussi and L. Stoica, $L^p$ estimates for the uniform norm of solutions of quasilinear SPDE's, Probab. Theory Related Fields, 133 (2005), 437–463. doi: 10.1007/s00440-005-0436-5. [8] K. Du and J. Liu, On the Cauchy problem for stochastic parabolic equations in Hölder spaces, Trans. Amer. Math. Soc., 371 (2019), 2643–2664. doi: 10.1090/tran/7533. [9] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493–540. doi: 10.1142/S0218202512500546. [10] E. P. Hsu, Y. Wang and Z. Wang, Stochastic De Giorgi iteration and regularity of stochastic partial differential equations, Ann. Probab., 45 (2017), 2855–2866. doi: 10.1214/16-AOP1126. [11] C. Imbert, A non-local regularization of first order Hamilton-Jacobi equations, J. Differential Equations, 211 (2005), 218–246. doi: 10.1016/j.jde.2004.06.001. [12] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferential functions further results, Comput. Math. Appl., 51 (2006), 1367–1376. doi: 10.1016/j.camwa.2006.02.001. [13] I. Kim, A BMO estimate for stochastic singular integral operators and its application to SPDEs, J. Funct. Anal., 269 (2015), 1289-1309.  doi: 10.1016/j.jfa.2015.05.015. [14] I. Kim and K.-H. Kim, An $L_p$-theory for stochastic partial differential equations driven by Lévy processes with pseudo-differential operators of arbitrary order, Stochastic Process. Appl., 126 (2016), 2761-2786.  doi: 10.1016/j.spa.2016.03.001. [15] I. Kim, K.-H. Kim and S. Lim, Parabolic Littlewood-Paley inequality for a class of time-dependent pseudo-differential operators of arbitrary order, and applications to high-order stochastic PDE, J. Math. Anal. Appl., 436 (2016), 1023-1047.  doi: 10.1016/j.jmaa.2015.12.040. [16] K.-H. Kim, $L_q (L_p)$ theory and Hölder estimates for parabolic SPDEs, Stochastic Process. Appl., 114 (2004), 313–330. doi: 10.1016/j.spa.2004.07.004. [17] K.-H. Kim and P. Kim, An $L_p$-theory of a class of stochastic equations with the random fractional Laplacian driven by Lévy processes, Stochastic Process. Appl., 122 (2012), 3921-3952.  doi: 10.1016/j.spa.2012.08.001. [18] K.-H. Kim, K. Lee and J. Seo, A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic in $\mathbb{R}^d$, J. Differential Equations, 291 (2021), 154-194.  doi: 10.1016/j.jde.2021.05.001. [19] K.-H. Kim, D. Park and J. Ryu, An $L_q(L_p)$-theory for diffusion equations with space-time nonlocal operators, J. Differential Equations, 287 (2021), 376-427.  doi: 10.1016/j.jde.2021.04.003. [20] N. V. Krylov, An analytic approach to SPDEs, in: Stochastic Partial Differential Equations: Six Perspectives, in: Math. Surveys Monogr., 64 (1999), 185-242. doi: 10.1090/surv/064/05. [21] N. V. Krylov, On $L_p$-theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal., 27 (1996), 313–340. doi: 10.1137/S0036141094263317. [22] S. B. Kuksin, N. S. Nadirashvili and A. L. Piatnitski, Hölder estimates for solutions of parabolic SPDEs, Theory Probab. Appl., 47 (2003), 157–164. doi: 10.1137/S0040585X97979524. [23] K. Li and J. Peng, Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941–959. doi: 10.1007/s00033-013-0369-2. [24] G. Lv, H. Gao, J. Wei and J.-L. Wu, BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666–2717. doi: 10.1016/j.jde.2018.08.042. [25] G. Lv, H. Gao, J. Wei and J.-L. Wu, Hölder estimates for solutions of stochastic nonlocal diffusion equations, Stochastic PDEs and Modelling of Multiscale Complex System, 97–110, Interdiscip. Math. Sci., 20, World Sci. Publ., Hackensack, NJ, 2019. [26] J. Neerven, M. C. Veraar and L. Weis, Maximal $L_p$-Regularity for Stochastic Evolution Equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. [27] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag Berlin, 1991. doi: 10.1007/978-3-662-21726-9. [28] R. Tian, L. Ding, J. Wei and S. Zheng, Hölder estimates of mild solutions for nonlocal SPDEs, Adv. Difference Equ., (2019), Paper No. 159, 12 pp. doi: 10.1186/s13662-019-2097-1. [29] J. van Neerven, M. Veraar and L. Weis, Stochastic maximal $L_p$regularity, Ann. Probab., 40 (2012), 788-812.  doi: 10.1214/10-AOP626. [30] X. Wang, Hölder continuous of the solutions to stochastic nonlocal heat equations, Comput. Math. Appl., 78 (2019), 741–753. doi: 10.1016/j.camwa.2019.02.036. [31] J. Wei, G. Lv and W. Wang, Stochastic transport equation with bounded and Dini continuous drift, J. Differential Equations, 323 (2022), 359–403. doi: 10.1016/j.jde.2022.03.038. [32] X. Xie, J. Duan, X. Li and G. Lv, A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift, arXiv: 1504.04631. [33] X. Zhang, $L_p$-theory of semi-linear SPDEs on general measure spaces and applications, J. Funct. Anal., 239 (2006), 44-75.  doi: 10.1016/j.jfa.2006.01.014. [34] X. Zhang, $L_p$-maximal regularity of nonlocal parabolic equations and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 573-614.  doi: 10.1016/j.anihpc.2012.10.006.
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