November  2016, 21(9): 3269-3299. doi: 10.3934/dcdsb.2016097

Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises

1. 

School of Science, Zhejiang University of Technology, No. 288, Liuhe Road, Xihu District, Hangzhou 310023, China

2. 

Department of Mathematics, University of York, Heslington Road, York YO10 5DD

Received  April 2016 Revised  April 2016 Published  October 2016

We study a class of abstract nonlinear stochastic equations of hyperbolic type driven by jump noises, which covers both beam equations with nonlocal, nonlinear terms and nonlinear wave equations. We derive an Itô formula for the local mild solution which plays an important role in the proof of our main results. Under appropriate conditions, we prove the non-explosion and the asymptotic stability of the mild solution.
Citation: 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
References:
[1]

Aït-Sahalia, Telling from discrete data whether the underlying continuous-time model is a diffusion,, Journal of Finance, 57 (2002), 2075.   Google Scholar

[2]

S. Albeverio, Z. Brzeźniak and J.-L. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients,, Journal of Mathematical Analysis and Applications, 371 (2010), 309.  doi: 10.1016/j.jmaa.2010.05.039.  Google Scholar

[3]

Z. Brzeźniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise,, Nonlinear Analysis: Real World Applications, 17 (2014), 283.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

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W. E. Baylis and J. Huschilt, Energy balance with the Landau-Lifshitz equation,, Phys. Lett. A, 301 (2002), 7.  doi: 10.1016/S0375-9601(02)00963-5.  Google Scholar

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D. Burgreen, Free Vibrations of a Pin-Ended Column with Constant Distance Between Pin Ends,, No. PIBAL-166. POLYTECHNIC INST OF BROOKLYN NY, (1950).   Google Scholar

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Z. Brzeźniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds,, Methods Funct. Anal. Topology, 6 (2000), 43.   Google Scholar

[7]

Z. Brzeźniak and D. Gątarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces,, Stochastic Processes and Their Applications, 84 (1999), 187.  doi: 10.1016/S0304-4149(99)00034-4.  Google Scholar

[8]

Z. Brzeźniak, B. Maslowski and J. Seidler, Stochastic nonlinear beam equations,, Probability Theory and Related Fields 132 (2005), 132 (2005), 119.  doi: 10.1007/s00440-004-0392-5.  Google Scholar

[9]

Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Martingale solutions for stochastic equation of reaction diffusion type driven by Lévy noise or Poisson random measure, preprint,, , ().   Google Scholar

[10]

J. F. Burrow, P. D. Baxter and J. W. Pitchford, Lévy processes, saltatory foraging, and superdiffusion,, Mathematical Modelling of Natural Phenomena 3 (2008), 3 (2008), 115.  doi: 10.1051/mmnp:2008060.  Google Scholar

[11]

A. Carroll, The Stochastic Nonlinear Heat Equation,, Ph. D. Thesis, (1999).   Google Scholar

[12]

P. L. Chow and J. L. Menaldi, Stochastic PDE for nonlinear vibration of elastic panels,, Differential Integral Equations, 12 (1999), 419.   Google Scholar

[13]

S. R. Das, The surprise element: Jumps in interest rates,, Journal of Econometrics, 106 (2002), 27.  doi: 10.1016/S0304-4076(01)00085-9.  Google Scholar

[14]

J. G. Eisley, Nonlinear vibration of beams and rectangular plates,, Zeitschrift für angewandte Mathematik und Physik ZAMP, 15 (1964), 167.  doi: 10.1007/BF01602658.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Reprint of the 1998 edition, (1998).   Google Scholar

[16]

I. Gyöngy and N. V. Krylov, On Stochastic Equations with Respect to Semimartingales. I,, Stochastics: An International Journal of Probability and Stochastic Processes, 4 (1980), 1.  doi: 10.1080/03610918008833154.  Google Scholar

[17]

I. Gyöngy, On stochastic equations with respect to semimartingale III,, Stochastics: An International Journal of Probability and Stochastic Processes, 7 (1982), 231.  doi: 10.1080/17442508208833220.  Google Scholar

[18]

E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure,, Electron. J. Probab, 10 (2005), 1496.  doi: 10.1214/EJP.v10-297.  Google Scholar

[19]

P. D. Lax and R. S. Phillips, Scattering Theory,, Pure and Applied Mathematics, (1967).   Google Scholar

[20]

R. Z. Khas'minskii, Stability of systems of differential equations under random perturbations of their parameters,, Izdat., (1969).   Google Scholar

[21]

B. Maslowski, J. Seidler and I. Vrkoč, Integral continuity and stability for stochastic hyperbolic equations,, Differential Integral Equations, 6 (1993), 355.   Google Scholar

[22]

M. Métivier, Semimartingales, A Course on Stochastic Processes,, de Gruyter Studies in Mathematics, (1982).   Google Scholar

[23]

M. Ondreját, a private communication to, [8], ().   Google Scholar

[24]

S. K. Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation,, Journal of Differential Equations, 135 (1997), 299.  doi: 10.1006/jdeq.1996.3231.  Google Scholar

[25]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach,, Encyclopedia of Mathematics and its Applications, (2007).  doi: 10.1017/CBO9780511721373.  Google Scholar

[26]

A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite-dimensional systems,, SIAM Review, 23 (1981), 25.  doi: 10.1137/1023003.  Google Scholar

[27]

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

[28]

T. Russo, P. Baldi, A. Parisi, G. Magnifico, S. Mariani and S. Cataudella, Lévy processes and stochastic von Bertalanffy models of growth, with application to fish population analysis,, Journal of Theoretical Biology, 258 (2009), 521.  doi: 10.1016/j.jtbi.2009.01.033.  Google Scholar

[29]

L. Tubaro, On abstract stochastic differential equation in Hilbert spaces with dissipative drift,, Stochastic Analysis and Applications, 1 (1983), 205.  doi: 10.1080/07362998308809012.  Google Scholar

[30]

L. Tubaro, An estimate of Burkholder type for stochastic processes defined by the stochastic integral,, Stochastic Analysis and Applications, 2 (1984), 187.  doi: 10.1080/07362998408809032.  Google Scholar

[31]

J. Van Neerven and J. Zhu, A maximal inequality for stochastic convolutions in 2-smooth Banach spaces,, Electron. Commun. Probab, 16 (2011), 689.  doi: 10.1214/ECP.v16-1677.  Google Scholar

[32]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Mathematical Library, (1978).   Google Scholar

[33]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17 (1950), 35.   Google Scholar

[34]

J. Zhu, A Study of SPDEs w.r.t. Compensated Poisson Random Measures and Related Topics,, Ph. D. Thesis, (2010).   Google Scholar

[35]

J. Zhu, Z. Brzeźniak and E. Hausenblas, Maximal inequality of stochastic convolution driven by compensated Poisson random measures in Banach spaces, preprint,, , ().   Google Scholar

show all references

References:
[1]

Aït-Sahalia, Telling from discrete data whether the underlying continuous-time model is a diffusion,, Journal of Finance, 57 (2002), 2075.   Google Scholar

[2]

S. Albeverio, Z. Brzeźniak and J.-L. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients,, Journal of Mathematical Analysis and Applications, 371 (2010), 309.  doi: 10.1016/j.jmaa.2010.05.039.  Google Scholar

[3]

Z. Brzeźniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise,, Nonlinear Analysis: Real World Applications, 17 (2014), 283.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

[4]

W. E. Baylis and J. Huschilt, Energy balance with the Landau-Lifshitz equation,, Phys. Lett. A, 301 (2002), 7.  doi: 10.1016/S0375-9601(02)00963-5.  Google Scholar

[5]

D. Burgreen, Free Vibrations of a Pin-Ended Column with Constant Distance Between Pin Ends,, No. PIBAL-166. POLYTECHNIC INST OF BROOKLYN NY, (1950).   Google Scholar

[6]

Z. Brzeźniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds,, Methods Funct. Anal. Topology, 6 (2000), 43.   Google Scholar

[7]

Z. Brzeźniak and D. Gątarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces,, Stochastic Processes and Their Applications, 84 (1999), 187.  doi: 10.1016/S0304-4149(99)00034-4.  Google Scholar

[8]

Z. Brzeźniak, B. Maslowski and J. Seidler, Stochastic nonlinear beam equations,, Probability Theory and Related Fields 132 (2005), 132 (2005), 119.  doi: 10.1007/s00440-004-0392-5.  Google Scholar

[9]

Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Martingale solutions for stochastic equation of reaction diffusion type driven by Lévy noise or Poisson random measure, preprint,, , ().   Google Scholar

[10]

J. F. Burrow, P. D. Baxter and J. W. Pitchford, Lévy processes, saltatory foraging, and superdiffusion,, Mathematical Modelling of Natural Phenomena 3 (2008), 3 (2008), 115.  doi: 10.1051/mmnp:2008060.  Google Scholar

[11]

A. Carroll, The Stochastic Nonlinear Heat Equation,, Ph. D. Thesis, (1999).   Google Scholar

[12]

P. L. Chow and J. L. Menaldi, Stochastic PDE for nonlinear vibration of elastic panels,, Differential Integral Equations, 12 (1999), 419.   Google Scholar

[13]

S. R. Das, The surprise element: Jumps in interest rates,, Journal of Econometrics, 106 (2002), 27.  doi: 10.1016/S0304-4076(01)00085-9.  Google Scholar

[14]

J. G. Eisley, Nonlinear vibration of beams and rectangular plates,, Zeitschrift für angewandte Mathematik und Physik ZAMP, 15 (1964), 167.  doi: 10.1007/BF01602658.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Reprint of the 1998 edition, (1998).   Google Scholar

[16]

I. Gyöngy and N. V. Krylov, On Stochastic Equations with Respect to Semimartingales. I,, Stochastics: An International Journal of Probability and Stochastic Processes, 4 (1980), 1.  doi: 10.1080/03610918008833154.  Google Scholar

[17]

I. Gyöngy, On stochastic equations with respect to semimartingale III,, Stochastics: An International Journal of Probability and Stochastic Processes, 7 (1982), 231.  doi: 10.1080/17442508208833220.  Google Scholar

[18]

E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure,, Electron. J. Probab, 10 (2005), 1496.  doi: 10.1214/EJP.v10-297.  Google Scholar

[19]

P. D. Lax and R. S. Phillips, Scattering Theory,, Pure and Applied Mathematics, (1967).   Google Scholar

[20]

R. Z. Khas'minskii, Stability of systems of differential equations under random perturbations of their parameters,, Izdat., (1969).   Google Scholar

[21]

B. Maslowski, J. Seidler and I. Vrkoč, Integral continuity and stability for stochastic hyperbolic equations,, Differential Integral Equations, 6 (1993), 355.   Google Scholar

[22]

M. Métivier, Semimartingales, A Course on Stochastic Processes,, de Gruyter Studies in Mathematics, (1982).   Google Scholar

[23]

M. Ondreját, a private communication to, [8], ().   Google Scholar

[24]

S. K. Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation,, Journal of Differential Equations, 135 (1997), 299.  doi: 10.1006/jdeq.1996.3231.  Google Scholar

[25]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach,, Encyclopedia of Mathematics and its Applications, (2007).  doi: 10.1017/CBO9780511721373.  Google Scholar

[26]

A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite-dimensional systems,, SIAM Review, 23 (1981), 25.  doi: 10.1137/1023003.  Google Scholar

[27]

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

[28]

T. Russo, P. Baldi, A. Parisi, G. Magnifico, S. Mariani and S. Cataudella, Lévy processes and stochastic von Bertalanffy models of growth, with application to fish population analysis,, Journal of Theoretical Biology, 258 (2009), 521.  doi: 10.1016/j.jtbi.2009.01.033.  Google Scholar

[29]

L. Tubaro, On abstract stochastic differential equation in Hilbert spaces with dissipative drift,, Stochastic Analysis and Applications, 1 (1983), 205.  doi: 10.1080/07362998308809012.  Google Scholar

[30]

L. Tubaro, An estimate of Burkholder type for stochastic processes defined by the stochastic integral,, Stochastic Analysis and Applications, 2 (1984), 187.  doi: 10.1080/07362998408809032.  Google Scholar

[31]

J. Van Neerven and J. Zhu, A maximal inequality for stochastic convolutions in 2-smooth Banach spaces,, Electron. Commun. Probab, 16 (2011), 689.  doi: 10.1214/ECP.v16-1677.  Google Scholar

[32]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Mathematical Library, (1978).   Google Scholar

[33]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17 (1950), 35.   Google Scholar

[34]

J. Zhu, A Study of SPDEs w.r.t. Compensated Poisson Random Measures and Related Topics,, Ph. D. Thesis, (2010).   Google Scholar

[35]

J. Zhu, Z. Brzeźniak and E. Hausenblas, Maximal inequality of stochastic convolution driven by compensated Poisson random measures in Banach spaces, preprint,, , ().   Google Scholar

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