Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

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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 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 and 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.
[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.
[3]	N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge: Cambridge University Press, 1989.
[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.
[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.
[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.
[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.
[8]	W. Feller, An Introduction to Probability Theory and Its Applications. Vol. Ⅱ, New York-London-Sydney: John Wiley & Sons, Inc., second edition, 1971.
[9]	I. Gyöngy, On stochastic equations with respect to semimartingales. Ⅲ, Stochastics, 7 (1982), 231-254. doi: 10.1080/17442508208833220.
[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.
[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.
[12]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Co., 1981.
[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.
[14]	T. Kosmala, Stochastic Partial Differential Equations Driven by Cylindrical Lévy Processes, PhD thesis, King's College London, 2020.
[15]	N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki, 14 (1979), 71–147.
[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.
[17]	U. Kumar and M. Riedle, The stochastic Cauchy problem driven by a cylindrical Lévy process, Electron. J. Probab., 25 (2020), paper no. 10.
[18]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Cham: Springer, 2015. doi: 10.1007/978-3-319-22354-4.
[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.
[20]	M. Métivier, Semimartingales: A Course on Stochastic Processes, Berlin: Walter de Gruyter & Co., 1982.
[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.
[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.
[23]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007.
[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.
[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.
[26]	M. Riedle, Infinitely divisible cylindrical measures on Banach spaces, Studia Math., 207 (2011), 235-256. doi: 10.4064/sm207-3-2.
[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.
[28]	M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Anal., 42 (2015), 809-838. doi: 10.1007/s11118-014-9458-x.
[29]	G. Samorodnitsky, Stochastic Processes and Long Range Dependence, Cham: Springer, 2016. doi: 10.1007/978-3-319-45575-4.
[30]	K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge: Cambridge University Press, 2013.

show all references

References:

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge: Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781.
[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.
[3]	N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge: Cambridge University Press, 1989.
[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.
[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.
[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.
[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.
[8]	W. Feller, An Introduction to Probability Theory and Its Applications. Vol. Ⅱ, New York-London-Sydney: John Wiley & Sons, Inc., second edition, 1971.
[9]	I. Gyöngy, On stochastic equations with respect to semimartingales. Ⅲ, Stochastics, 7 (1982), 231-254. doi: 10.1080/17442508208833220.
[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.
[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.
[12]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Co., 1981.
[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.
[14]	T. Kosmala, Stochastic Partial Differential Equations Driven by Cylindrical Lévy Processes, PhD thesis, King's College London, 2020.
[15]	N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki, 14 (1979), 71–147.
[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.
[17]	U. Kumar and M. Riedle, The stochastic Cauchy problem driven by a cylindrical Lévy process, Electron. J. Probab., 25 (2020), paper no. 10.
[18]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Cham: Springer, 2015. doi: 10.1007/978-3-319-22354-4.
[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.
[20]	M. Métivier, Semimartingales: A Course on Stochastic Processes, Berlin: Walter de Gruyter & Co., 1982.
[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.
[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.
[23]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007.
[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.
[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.
[26]	M. Riedle, Infinitely divisible cylindrical measures on Banach spaces, Studia Math., 207 (2011), 235-256. doi: 10.4064/sm207-3-2.
[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.
[28]	M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Anal., 42 (2015), 809-838. doi: 10.1007/s11118-014-9458-x.
[29]	G. Samorodnitsky, Stochastic Processes and Long Range Dependence, Cham: Springer, 2016. doi: 10.1007/978-3-319-45575-4.
[30]	K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge: Cambridge University Press, 2013.

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