# American Institute of Mathematical Sciences

• Previous Article
Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
• DCDS-B Home
• This Issue
• Next Article
$L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity
July  2019, 24(7): 2989-3009. doi: 10.3934/dcdsb.2018296

## Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises

 1 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, No. 30 Xueyuan Road, Haidian, Beijing 100083, China 2 Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan

* Corresponding author: Bin Xie

Received  January 2018 Revised  June 2018 Published  October 2018

Fund Project: The first author is supported in part by NSF of China (No.11571030) and the second author is supported by JSPS KAKENH (No.16K05197).

Two properties of stochastic heat equations driven by impulsive noises, which are also called Lévy space-time white noises, are mainly investigated in this paper. We first study the comparison theorem for two stochastic heat equations driven by same noises under some sufficient condition, which is proved via the application of Itô's formula. In particular, we obtain the non-negativity of solutions with non-negative initial data. Then, we investigate the positive correlation of the solutions as the application of the comparison theorem. We prove that the total masses of two solutions relative to two different stochastic heat equations with same noise become nearly uncorrelated after a long time.

Citation: Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
##### References:
 [1] S. Albeverio, J.-L. Wu and T.-S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 74 (1998), 21-36.  doi: 10.1016/S0304-4149(97)00112-9.  Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [3] Z. Brzeźniak, W. Liu and J.-H. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar [4] Z. Brzeźniak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal., 32 (2010), 153-188.  doi: 10.1007/s11118-009-9149-1.  Google Scholar [5] L. Chen, D. Khoshnevisan and K. Kim, Decorrelation of total mass via energy, Potential Anal., 45 (2016), 157-166.  doi: 10.1007/s11118-016-9540-7.  Google Scholar [6] H. Dadashi, Large deviation principle for semilinear stochastic evolution equations with Poisson noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20 (2017), 1750009, 29 pp. doi: 10.1142/S0219025717500096.  Google Scholar [7] K. A. Dareiotis and I. Gyöngy, A comparison principle for stochastic integro-differential equations, Potential Anal., 41 (2014), 1203-1222.  doi: 10.1007/s11118-014-9416-7.  Google Scholar [8] C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Relat. Fields, 95 (1993), 1-24.  doi: 10.1007/BF01197335.  Google Scholar [9] Z. Dong, L. H. Xu and X. C. Zhang, Exponential ergodicity of stochastic Burgers equations driven by α-stable processes, J. Stat. Phys., 154 (2014), 929-949.  doi: 10.1007/s10955-013-0881-y.  Google Scholar [10] Z. Dong, J. Xiong, J. L. Zhai and T. S. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254.  doi: 10.1016/j.jfa.2016.10.012.  Google Scholar [11] M. Foondun and D. Khoshnevisan, Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab., 14 (2009), 548-568.  doi: 10.1214/EJP.v14-614.  Google Scholar [12] M. Foondun and E. Nualart, On the behaviour of stochastic heat equations on bounded domains, ALEA Lat. Am. J. Probab. Math. Stat., 12 (2015), 551-571.   Google Scholar [13] T. Funaki and S. Olla, Fluctuations for $\nabla \phi$ interface model on a wall, Stochastic Process. Appl., 94 (2001), 1-27.  doi: 10.1016/S0304-4149(00)00104-6.  Google Scholar [14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd Edition, North-Holland, Kodansha, 1989.  Google Scholar [15] I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [16] P. Kotelenez, Comparison methods for a class of function valued stochastic partial differential equations, Probab. Theory Relat. Fields, 93 (1992), 1-19.  doi: 10.1007/BF01195385.  Google Scholar [17] C. Marinelli and M. Röckner, Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise, Electron. J. Probab., 15 (2010), 1528-1555.  doi: 10.1214/EJP.v15-818.  Google Scholar [18] C. Mueller, On the support of solutions to the heat equation with noise, Stochastics Stochastics Rep., 37 (1991), 225-245.  doi: 10.1080/17442509108833738.  Google Scholar [19] C. Mueller and D. Nualart, Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008), 2248-2258.  doi: 10.1214/EJP.v13-589.  Google Scholar [20] S. G. Peng and X. H. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.  doi: 10.1016/j.spa.2005.08.004.  Google Scholar [21] S. Peszat and J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise, Stochastic Partial Differential Equations and Applications-VII, Lect. Notes Pure Appl. Math., 245 (2006), 229-242. doi: 10.1201/9781420028720.ch19.  Google Scholar [22] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, 113, 2007. xii+419 pp. doi: 10.1017/CBO9780511721373.  Google Scholar [23] T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Canad. J. Math., 46 (1994), 415-437.  doi: 10.4153/CJM-1994-022-8.  Google Scholar [24] Y.-L. Song and T.-G. Xu, Exponential convergence for some SPDEs with Lévy noises, Illinois J. Math., 60 (2016), 587-611.   Google Scholar [25] J. B. Walsh, An introduction to stochastic partial differential equations, Ecole d' Eté de Probabilités de Saint-Flour, XIV-1984, Lect. Notes Math., 1180, Springer, Berlin, (1986), 265–439. doi: 10.1007/BFb0074920.  Google Scholar [26] J.-L. Wu and B. Xie, On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise in $\mathbb{R}^d$, Bull. Sci. Math., 136 (2012), 484-506.  doi: 10.1016/j.bulsci.2011.07.015.  Google Scholar [27] B. Xie, Impulsive noise driven one-dimensional higher-order fractional partial differential equations, Stoch. Anal. Appl., 30 (2012), 122-145.  doi: 10.1080/07362994.2012.628917.  Google Scholar [28] J. H. Zhu and Z. Brzeźniak, Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3269-3299.  doi: 10.3934/dcdsb.2016097.  Google Scholar

show all references

##### References:
 [1] S. Albeverio, J.-L. Wu and T.-S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 74 (1998), 21-36.  doi: 10.1016/S0304-4149(97)00112-9.  Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [3] Z. Brzeźniak, W. Liu and J.-H. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar [4] Z. Brzeźniak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal., 32 (2010), 153-188.  doi: 10.1007/s11118-009-9149-1.  Google Scholar [5] L. Chen, D. Khoshnevisan and K. Kim, Decorrelation of total mass via energy, Potential Anal., 45 (2016), 157-166.  doi: 10.1007/s11118-016-9540-7.  Google Scholar [6] H. Dadashi, Large deviation principle for semilinear stochastic evolution equations with Poisson noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20 (2017), 1750009, 29 pp. doi: 10.1142/S0219025717500096.  Google Scholar [7] K. A. Dareiotis and I. Gyöngy, A comparison principle for stochastic integro-differential equations, Potential Anal., 41 (2014), 1203-1222.  doi: 10.1007/s11118-014-9416-7.  Google Scholar [8] C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Relat. Fields, 95 (1993), 1-24.  doi: 10.1007/BF01197335.  Google Scholar [9] Z. Dong, L. H. Xu and X. C. Zhang, Exponential ergodicity of stochastic Burgers equations driven by α-stable processes, J. Stat. Phys., 154 (2014), 929-949.  doi: 10.1007/s10955-013-0881-y.  Google Scholar [10] Z. Dong, J. Xiong, J. L. Zhai and T. S. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254.  doi: 10.1016/j.jfa.2016.10.012.  Google Scholar [11] M. Foondun and D. Khoshnevisan, Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab., 14 (2009), 548-568.  doi: 10.1214/EJP.v14-614.  Google Scholar [12] M. Foondun and E. Nualart, On the behaviour of stochastic heat equations on bounded domains, ALEA Lat. Am. J. Probab. Math. Stat., 12 (2015), 551-571.   Google Scholar [13] T. Funaki and S. Olla, Fluctuations for $\nabla \phi$ interface model on a wall, Stochastic Process. Appl., 94 (2001), 1-27.  doi: 10.1016/S0304-4149(00)00104-6.  Google Scholar [14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd Edition, North-Holland, Kodansha, 1989.  Google Scholar [15] I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [16] P. Kotelenez, Comparison methods for a class of function valued stochastic partial differential equations, Probab. Theory Relat. Fields, 93 (1992), 1-19.  doi: 10.1007/BF01195385.  Google Scholar [17] C. Marinelli and M. Röckner, Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise, Electron. J. Probab., 15 (2010), 1528-1555.  doi: 10.1214/EJP.v15-818.  Google Scholar [18] C. Mueller, On the support of solutions to the heat equation with noise, Stochastics Stochastics Rep., 37 (1991), 225-245.  doi: 10.1080/17442509108833738.  Google Scholar [19] C. Mueller and D. Nualart, Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008), 2248-2258.  doi: 10.1214/EJP.v13-589.  Google Scholar [20] S. G. Peng and X. H. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.  doi: 10.1016/j.spa.2005.08.004.  Google Scholar [21] S. Peszat and J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise, Stochastic Partial Differential Equations and Applications-VII, Lect. Notes Pure Appl. Math., 245 (2006), 229-242. doi: 10.1201/9781420028720.ch19.  Google Scholar [22] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, 113, 2007. xii+419 pp. doi: 10.1017/CBO9780511721373.  Google Scholar [23] T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Canad. J. Math., 46 (1994), 415-437.  doi: 10.4153/CJM-1994-022-8.  Google Scholar [24] Y.-L. Song and T.-G. Xu, Exponential convergence for some SPDEs with Lévy noises, Illinois J. Math., 60 (2016), 587-611.   Google Scholar [25] J. B. Walsh, An introduction to stochastic partial differential equations, Ecole d' Eté de Probabilités de Saint-Flour, XIV-1984, Lect. Notes Math., 1180, Springer, Berlin, (1986), 265–439. doi: 10.1007/BFb0074920.  Google Scholar [26] J.-L. Wu and B. Xie, On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise in $\mathbb{R}^d$, Bull. Sci. Math., 136 (2012), 484-506.  doi: 10.1016/j.bulsci.2011.07.015.  Google Scholar [27] B. Xie, Impulsive noise driven one-dimensional higher-order fractional partial differential equations, Stoch. Anal. Appl., 30 (2012), 122-145.  doi: 10.1080/07362994.2012.628917.  Google Scholar [28] J. H. Zhu and Z. Brzeźniak, Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3269-3299.  doi: 10.3934/dcdsb.2016097.  Google Scholar
 [1] Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems & Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062 [2] Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374 [3] Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175 [4] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [5] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [6] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [7] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [8] Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009 [9] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [10] Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365 [11] Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020388 [12] Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 [13] Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $\mathbb{R}^{N}$ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376 [14] Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 [15] Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052 [16] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [17] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [18] Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 [19] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 [20] Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

2019 Impact Factor: 1.27

Article outline