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, 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.

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd Edition, North-Holland, Kodansha, 1989.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

Y.-L. Song and T.-G. Xu, Exponential convergence for some SPDEs with Lévy noises, Illinois J. Math., 60 (2016), 587-611.

[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.

[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.

[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.

[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.

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.

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd Edition, North-Holland, Kodansha, 1989.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

Y.-L. Song and T.-G. Xu, Exponential convergence for some SPDEs with Lévy noises, Illinois J. Math., 60 (2016), 587-611.

[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.

[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.

[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.

[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.

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