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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. AlbeverioJ.-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. BrzeźniakW. 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. ChenD. 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

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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. DongL. 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. DongJ. XiongJ. 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. AlbeverioJ.-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. BrzeźniakW. 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. ChenD. 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. DongL. 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. DongJ. XiongJ. 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

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