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.

[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.

[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.

[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.

[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).

[6]

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

[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.

[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.

[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,, , ().

[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.

[11]

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

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[35]

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

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.

[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.

[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.

[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.

[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).

[6]

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

[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.

[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.

[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,, , ().

[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.

[11]

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

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[35]

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

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