# American Institute of Mathematical Sciences

October  2020, 40(10): 5639-5710. doi: 10.3934/dcds.2020241

## Review of local and global existence results for stochastic pdes with Lévy noise

 1 Department of Mathematics, Indiana University, Swain East, Bloomington, IN 47405, USA 2 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA

* Corresponding author

Received  April 2019 Published  June 2020

Fund Project: The authors are supported by NSF grant DMS-1510249

This article is a review of Lévy processes, stochastic integration and existence results for stochastic differential equations and stochastic partial differential equations driven by Lévy noise. An abstract PDE of the typical type encountered in fluid mechanics is considered in a stochastic setting driven by a general Lévy noise. Existence and uniqueness of a local pathwise solution is established as a demonstration of general techniques in the area.

Citation: 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 - A, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241
##### References:
 [1] D. Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335-340.  doi: 10.1214/aop/1176995579.  Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, volume 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge University Press, Cambridge, 2009. Google Scholar [3] A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.  Google Scholar [4] P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edition, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar [5] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), 131-201.  doi: 10.1007/s11118-017-9651-9.  Google Scholar [6] 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 [7] J. Cyr, P. Nguyen and R. Temam, Stochastic one layer shallow water equations with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3765-3818.  doi: 10.3934/dcdsb.2018331.  Google Scholar [8] J. Cyr, S. Tang and R. Temam, A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, in Trends in Applications of Mathematics to Mechanics (eds. E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin), Springer INdAM Series, (2018), 289–373. doi: 10.1007/978-3-319-75940-1_14.  Google Scholar [9] J. Cyr, S. Tang and R. Temam, The Euler equations of an inviscid incompressible fluid driven by a Lévy noise, Nonlinear Anal. Real World Appl., 44 (2018), 173-222.  doi: 10.1016/j.nonrwa.2018.04.002.  Google Scholar [10] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. Ⅱ: General Theory and Structure, Probability and its Applications, $2^nd$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5.  Google Scholar [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 152 of Encyclopedia of Mathematics and its Applications, 2nd edition, Cambridge University Press, Cambridge, 2014. Google Scholar [12] A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar [13] A. Debussche, M. Högele and P. Imkeller, The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise, volume 2085 of Lecture Notes in Mathematics, Springer, Cham, 2013. doi: 10.1007/978-3-319-00828-8.  Google Scholar [14] P. D. Ditlevsen, Observation of α-stable noise induced millennial climate changes from an ice-core record, Geophys. Res. Lett., 26 (1999), 1441-1444.   Google Scholar [15] S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.  Google Scholar [16] F. Flandoli, An introduction to 3D stochastic fluid dynamics, in SPDE in Hydrodynamic: Recent Progress and Prospects, volume 1942 of Lecture Notes in Math., Springer, Berlin, (2008), 51–150. doi: 10.1007/978-3-540-78493-7_2.  Google Scholar [17] F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar [18] N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar [19] I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.  Google Scholar [20] N. Ikeda, M. Nagasawa and S. Watanabe, A construction of Markov processes by piecing out, Pro. Japan Acad., 42 (1966), 370-375.  doi: 10.3792/pja/1195522037.  Google Scholar [21] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, volume 24 of North-Holland Mathematical Library, 2nd edition, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar [22] O. Kallenberg, Foundations of Modern Probability, Probability and its Applications, Springer-Verlag, New York, 1997.  Google Scholar [23] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth.  Google Scholar [24] A. J. Majda and N. Chen, Simple dynamical models capturing the key features of the Central Pacific El Niño, P. Natl. Acad. Sci. USA, 113 (2016), 11732-11737.   Google Scholar [25] C. Marinelli and M. Röckner, On the maximal inequalities of Burkholder, Davis and Gundy, Expo. Math., 34 (2016), 1-26.  doi: 10.1016/j.exmath.2015.01.002.  Google Scholar [26] M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.  Google Scholar [27] M. Métivier, Stochastic Partial Differential Equations in Infinite-dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa], Scuola Normale Superiore, Pisa, 1988.  Google Scholar [28] M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptot. Anal., 99 (2016), 67-103.  doi: 10.3233/ASY-161376.  Google Scholar [29] E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3{D} domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.  Google Scholar [30] P. Nguyen, K. Tawri and R. Temam, Monotone operator equations of the Ladyzhenskaya-Smagorinsky type driven by a Lévy noise, in preparation. Google Scholar [31] C. Penland and B. D. Ewald, On modelling physical systems with stochastic models: diffusion versus Lévy processes, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 2457-2476.   Google Scholar [32] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise: An Evolution Equation Approach, volume 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. Google Scholar [33] M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of Numerical Analysis. Vol. XIV. Special Volume: Computational Methods for the Atmosphere and the Oceans, volume 14 of Handb. Numer. Anal., Elsevier/North-Holland, Amsterdam, (2009), 577–750. doi: 10.1016/S1570-8659(08)00212-3.  Google Scholar [34] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905 of Lecture Notes in Mathematics, Springer, Berlin, 2007.  Google Scholar [35] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 1999. Google Scholar [36] S. N. Stechmann and J. D. Neelin, First-passage-time prototypes for precipitation statistics, J. Atmos. Sci., 71 (2014), 3269-3291.  doi: 10.1175/JAS-D-13-0268.1.  Google Scholar [37] K. Tawri and R. Temam, Hilbertian Approximations of Monotone Operators, Pure and Applied Functional Analysis Google Scholar [38] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, first edition, 1983; second edition, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [39] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. Google Scholar [40] S. Thual, A. J. Majda, N. Chen and S. N. Stechmann, Simple stochastic model for El Niño with westerly wind bursts, Proc. Natl. Acad. Sci. USA, 113 (2016), 10245-10250.  doi: 10.1073/pnas.1612002113.  Google Scholar [41] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.  doi: 10.1215/kjm/1250523691.  Google Scholar

show all references

##### References:
 [1] D. Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335-340.  doi: 10.1214/aop/1176995579.  Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, volume 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge University Press, Cambridge, 2009. Google Scholar [3] A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.  Google Scholar [4] P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edition, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar [5] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), 131-201.  doi: 10.1007/s11118-017-9651-9.  Google Scholar [6] 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 [7] J. Cyr, P. Nguyen and R. Temam, Stochastic one layer shallow water equations with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3765-3818.  doi: 10.3934/dcdsb.2018331.  Google Scholar [8] J. Cyr, S. Tang and R. Temam, A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, in Trends in Applications of Mathematics to Mechanics (eds. E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin), Springer INdAM Series, (2018), 289–373. doi: 10.1007/978-3-319-75940-1_14.  Google Scholar [9] J. Cyr, S. Tang and R. Temam, The Euler equations of an inviscid incompressible fluid driven by a Lévy noise, Nonlinear Anal. Real World Appl., 44 (2018), 173-222.  doi: 10.1016/j.nonrwa.2018.04.002.  Google Scholar [10] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. Ⅱ: General Theory and Structure, Probability and its Applications, $2^nd$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5.  Google Scholar [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 152 of Encyclopedia of Mathematics and its Applications, 2nd edition, Cambridge University Press, Cambridge, 2014. Google Scholar [12] A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar [13] A. Debussche, M. Högele and P. Imkeller, The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise, volume 2085 of Lecture Notes in Mathematics, Springer, Cham, 2013. doi: 10.1007/978-3-319-00828-8.  Google Scholar [14] P. D. Ditlevsen, Observation of α-stable noise induced millennial climate changes from an ice-core record, Geophys. Res. Lett., 26 (1999), 1441-1444.   Google Scholar [15] S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.  Google Scholar [16] F. Flandoli, An introduction to 3D stochastic fluid dynamics, in SPDE in Hydrodynamic: Recent Progress and Prospects, volume 1942 of Lecture Notes in Math., Springer, Berlin, (2008), 51–150. doi: 10.1007/978-3-540-78493-7_2.  Google Scholar [17] F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar [18] N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar [19] I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.  Google Scholar [20] N. Ikeda, M. Nagasawa and S. Watanabe, A construction of Markov processes by piecing out, Pro. Japan Acad., 42 (1966), 370-375.  doi: 10.3792/pja/1195522037.  Google Scholar [21] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, volume 24 of North-Holland Mathematical Library, 2nd edition, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar [22] O. Kallenberg, Foundations of Modern Probability, Probability and its Applications, Springer-Verlag, New York, 1997.  Google Scholar [23] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth.  Google Scholar [24] A. J. Majda and N. Chen, Simple dynamical models capturing the key features of the Central Pacific El Niño, P. Natl. Acad. Sci. USA, 113 (2016), 11732-11737.   Google Scholar [25] C. Marinelli and M. Röckner, On the maximal inequalities of Burkholder, Davis and Gundy, Expo. Math., 34 (2016), 1-26.  doi: 10.1016/j.exmath.2015.01.002.  Google Scholar [26] M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.  Google Scholar [27] M. Métivier, Stochastic Partial Differential Equations in Infinite-dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa], Scuola Normale Superiore, Pisa, 1988.  Google Scholar [28] M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptot. Anal., 99 (2016), 67-103.  doi: 10.3233/ASY-161376.  Google Scholar [29] E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3{D} domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.  Google Scholar [30] P. Nguyen, K. Tawri and R. Temam, Monotone operator equations of the Ladyzhenskaya-Smagorinsky type driven by a Lévy noise, in preparation. Google Scholar [31] C. Penland and B. D. Ewald, On modelling physical systems with stochastic models: diffusion versus Lévy processes, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 2457-2476.   Google Scholar [32] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise: An Evolution Equation Approach, volume 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. Google Scholar [33] M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of Numerical Analysis. Vol. XIV. Special Volume: Computational Methods for the Atmosphere and the Oceans, volume 14 of Handb. Numer. Anal., Elsevier/North-Holland, Amsterdam, (2009), 577–750. doi: 10.1016/S1570-8659(08)00212-3.  Google Scholar [34] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905 of Lecture Notes in Mathematics, Springer, Berlin, 2007.  Google Scholar [35] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 1999. Google Scholar [36] S. N. Stechmann and J. D. Neelin, First-passage-time prototypes for precipitation statistics, J. Atmos. Sci., 71 (2014), 3269-3291.  doi: 10.1175/JAS-D-13-0268.1.  Google Scholar [37] K. Tawri and R. Temam, Hilbertian Approximations of Monotone Operators, Pure and Applied Functional Analysis Google Scholar [38] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, first edition, 1983; second edition, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [39] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. Google Scholar [40] S. Thual, A. J. Majda, N. Chen and S. N. Stechmann, Simple stochastic model for El Niño with westerly wind bursts, Proc. Natl. Acad. Sci. USA, 113 (2016), 10245-10250.  doi: 10.1073/pnas.1612002113.  Google Scholar [41] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.  doi: 10.1215/kjm/1250523691.  Google Scholar
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