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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  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
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