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## Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

 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.
Citation: Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 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

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##### 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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