American Institute of Mathematical Sciences

Advanced
June  2021, 26(6): 2879-2898. doi: 10.3934/dcdsb.2020209

Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

Tomasz Kosmala 1, and  Markus Riedle 2,3,,

1. 

School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK
2. 

Department of Mathematics, King's College London, London WC2R 2LS, UK
3. 

Institute of Mathematical Stochastics, Faculty of Mathematics, TU Dresden, 01062 Dresden, Germany

* Corresponding author: Markus Riedle

Received  December 2019 Revised  April 2020 Published  June 2021 Early access  July 2020

In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation
$ \, \mathrm{d}X(t) = F(X(t)) \, \mathrm{d}t + G(X(t)) \, \mathrm{d}L(t) $
driven by a cylindrical Lévy process
$ L $
 is established. The coefficients
$ F $
 and
$ G $
 are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.
Keywords: cylindrical Lévy processes, stochastic partial differential equations, multiplicative noise, variational solutions, stochastic integration.
Mathematics Subject Classification: Primary: 60H15; Secondary: 60G51, 60G20, 28A35.
Citation: Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge: Cambridge University Press, 2009.  doi: 10.1017/CBO9780511809781.  Google Scholar
[2]

D. Applebaum and M. Riedle, Cylindrical Lévy processes in Banach spaces, Proc. Lond. Math. Soc., 101 (2010), 697-726.  doi: 10.1112/plms/pdq004.  Google Scholar

[3] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge: Cambridge University Press, 1989.   Google Scholar
[4]

Z. Brzeźniak, B. Goldys, P. Imkeller, S. Peszat, E. Priola and J. Zabczyk, Time irregularity of generalized Ornstein-Uhlenbeck processes, C. R. Math. Acad. Sci. Paris, 348 (2010), 273-276.  doi: 10.1016/j.crma.2010.01.022.  Google Scholar

[5]

Z. Brzeźniak, W. Liu and J. 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

[6]

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

[7]

P. Embrechts and C. M. Goldie, Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure, Ann. Probab., 9 (1981), 468-481.  doi: 10.1214/aop/1176994419.  Google Scholar

[8]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. Ⅱ, New York-London-Sydney: John Wiley & Sons, Inc., second edition, 1971.  Google Scholar

[9]

I. Gyöngy, On stochastic equations with respect to semimartingales. Ⅲ, Stochastics, 7 (1982), 231-254.  doi: 10.1080/17442508208833220.  Google Scholar

[10]

I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics, 4 (1980/81), 1-21.  doi: 10.1080/03610918008833154.  Google Scholar

[11]

I. Gyöngy and N. V. Krylov, On stochastics equations with respect to semimartingales. Ⅱ. Itô formula in Banach spaces, Stochastics, 6 (1981/82), 153-173.  doi: 10.1080/17442508208833202.  Google Scholar

[12]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Co., 1981.  Google Scholar

[13]

A. Jakubowski and M. Riedle, Stochastic integration with respect to cylindrical Lévy processes, Ann. Probab., 45 (2017), 4273-4306.  doi: 10.1214/16-AOP1164.  Google Scholar

[14]

T. Kosmala, Stochastic Partial Differential Equations Driven by Cylindrical Lévy Processes, PhD thesis, King's College London, 2020. Google Scholar

[15]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki, 14 (1979), 71–147.  Google Scholar

[16]

U. Kumar and M. Riedle, Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process, 2019., Preprint available at https://arXiv.org/abs/1904.03118. Google Scholar

[17]

U. Kumar and M. Riedle, The stochastic Cauchy problem driven by a cylindrical Lévy process, Electron. J. Probab., 25 (2020), paper no. 10. Google Scholar

[18]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Cham: Springer, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[19]

Y. Liu and J. Zhai, A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise, C. R. Math. Acad. Sci. Paris, 350 (2012), 97-100.  doi: 10.1016/j.crma.2011.11.017.  Google Scholar

[20]

M. Métivier, Semimartingales: A Course on Stochastic Processes, Berlin: Walter de Gruyter & Co., 1982.  Google Scholar

[21] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Cambridge: Cambridge University Press, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[22]

S. Peszat and J. Zabczyk, Time regularity of solutions to linear equations with Lévy noise in infinite dimensions, Stochastic Process. Appl., 123 (2013), 719-751.  doi: 10.1016/j.spa.2012.10.012.  Google Scholar

[23]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007.  Google Scholar

[24]

E. Priola and J. Zabczyk, On linear evolution equations for a class of cylindrical Lévy noises, In Stochastic Partial Differential Equations and Applications, volume 25 of Quad. Mat., 223–242. Dept. Math., Seconda Univ. Napoli, Caserta, 2010.  Google Scholar

[25]

E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

[26]

M. Riedle, Infinitely divisible cylindrical measures on Banach spaces, Studia Math., 207 (2011), 235-256.  doi: 10.4064/sm207-3-2.  Google Scholar

[27]

M. Riedle, Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: an $L^2$ approach, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 1450008, 19 pp. doi: 10.1142/S0219025714500088.  Google Scholar

[28]

M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Anal., 42 (2015), 809-838.  doi: 10.1007/s11118-014-9458-x.  Google Scholar

[29]

G. Samorodnitsky, Stochastic Processes and Long Range Dependence, Cham: Springer, 2016. doi: 10.1007/978-3-319-45575-4.  Google Scholar

[30] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge: Cambridge University Press, 2013.   Google Scholar

show all references

References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge: Cambridge University Press, 2009.  doi: 10.1017/CBO9780511809781.  Google Scholar
[2]

D. Applebaum and M. Riedle, Cylindrical Lévy processes in Banach spaces, Proc. Lond. Math. Soc., 101 (2010), 697-726.  doi: 10.1112/plms/pdq004.  Google Scholar

[3] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge: Cambridge University Press, 1989.   Google Scholar
[4]

Z. Brzeźniak, B. Goldys, P. Imkeller, S. Peszat, E. Priola and J. Zabczyk, Time irregularity of generalized Ornstein-Uhlenbeck processes, C. R. Math. Acad. Sci. Paris, 348 (2010), 273-276.  doi: 10.1016/j.crma.2010.01.022.  Google Scholar

[5]

Z. Brzeźniak, W. Liu and J. 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

[6]

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

[7]

P. Embrechts and C. M. Goldie, Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure, Ann. Probab., 9 (1981), 468-481.  doi: 10.1214/aop/1176994419.  Google Scholar

[8]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. Ⅱ, New York-London-Sydney: John Wiley & Sons, Inc., second edition, 1971.  Google Scholar

[9]

I. Gyöngy, On stochastic equations with respect to semimartingales. Ⅲ, Stochastics, 7 (1982), 231-254.  doi: 10.1080/17442508208833220.  Google Scholar

[10]

I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics, 4 (1980/81), 1-21.  doi: 10.1080/03610918008833154.  Google Scholar

[11]

I. Gyöngy and N. V. Krylov, On stochastics equations with respect to semimartingales. Ⅱ. Itô formula in Banach spaces, Stochastics, 6 (1981/82), 153-173.  doi: 10.1080/17442508208833202.  Google Scholar

[12]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Co., 1981.  Google Scholar

[13]

A. Jakubowski and M. Riedle, Stochastic integration with respect to cylindrical Lévy processes, Ann. Probab., 45 (2017), 4273-4306.  doi: 10.1214/16-AOP1164.  Google Scholar

[14]

T. Kosmala, Stochastic Partial Differential Equations Driven by Cylindrical Lévy Processes, PhD thesis, King's College London, 2020. Google Scholar

[15]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki, 14 (1979), 71–147.  Google Scholar

[16]

U. Kumar and M. Riedle, Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process, 2019., Preprint available at https://arXiv.org/abs/1904.03118. Google Scholar

[17]

U. Kumar and M. Riedle, The stochastic Cauchy problem driven by a cylindrical Lévy process, Electron. J. Probab., 25 (2020), paper no. 10. Google Scholar

[18]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Cham: Springer, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[19]

Y. Liu and J. Zhai, A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise, C. R. Math. Acad. Sci. Paris, 350 (2012), 97-100.  doi: 10.1016/j.crma.2011.11.017.  Google Scholar

[20]

M. Métivier, Semimartingales: A Course on Stochastic Processes, Berlin: Walter de Gruyter & Co., 1982.  Google Scholar

[21] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Cambridge: Cambridge University Press, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[22]

S. Peszat and J. Zabczyk, Time regularity of solutions to linear equations with Lévy noise in infinite dimensions, Stochastic Process. Appl., 123 (2013), 719-751.  doi: 10.1016/j.spa.2012.10.012.  Google Scholar

[23]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007.  Google Scholar

[24]

E. Priola and J. Zabczyk, On linear evolution equations for a class of cylindrical Lévy noises, In Stochastic Partial Differential Equations and Applications, volume 25 of Quad. Mat., 223–242. Dept. Math., Seconda Univ. Napoli, Caserta, 2010.  Google Scholar

[25]

E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

[26]

M. Riedle, Infinitely divisible cylindrical measures on Banach spaces, Studia Math., 207 (2011), 235-256.  doi: 10.4064/sm207-3-2.  Google Scholar

[27]

M. Riedle, Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: an $L^2$ approach, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 1450008, 19 pp. doi: 10.1142/S0219025714500088.  Google Scholar

[28]

M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Anal., 42 (2015), 809-838.  doi: 10.1007/s11118-014-9458-x.  Google Scholar

[29]

G. Samorodnitsky, Stochastic Processes and Long Range Dependence, Cham: Springer, 2016. doi: 10.1007/978-3-319-45575-4.  Google Scholar

[30] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge: Cambridge University Press, 2013.   Google Scholar
[1]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[2]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[3]

Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021207
[4]

Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021204
[5]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[6]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331
[7]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[8]

Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282
[9]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
[10]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[11]

Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021
[12]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[13]

Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080
[14]

Justin Cyr, Phuong Nguyen, Sisi Tang, Roger Temam. Review of local and global existence results for stochastic pdes with Lévy noise. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241
[15]

Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137
[16]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[17]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047
[18]

Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087
[19]

T. Tachim Medjo. The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 177-197. doi: 10.3934/dcdsb.2010.14.177
[20]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (215)
  • HTML views (330)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences