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Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. (English) Zbl 1378.60093
The authors investigate the existence and uniqueness of global mild solutions to the abstract second-order stochastic differential equation perturbed by jump noises \[ u_{tt} + A^2 u + \nabla_u \Phi(u) + f(t, u, u_t) = \int_Z g(t,u,u_t, z) \tilde{N}(t, dz), \tag{1} \] where \(u\) is a Hilbert space \(H\)-valued stochastic process, \(A\) is a self-adjoint operator on \(H\) such that \(A \geq \mu I\) for some \(\mu > 0\), \(\Phi\) : \(D(A)\) \(\to\) \([0, \infty)\) is an energy function and \(\tilde{N}\) is a compensated Poisson random measure. Actually, this equation covers both beam equations with nonlocal nonlinear terms and nonlinear wave equations. The proof for the existence of global solutions is technically due to an Itô formula for the local mild solution. Under appropriate conditions on \(f\) and \(g\), they prove the non-explosion and the asymptotic stability of the mild solution. In particular, the unique global mild solution \(w\) \(=\) \((u, u_t)\) satisfies the following estimate \[ \sup_{ t \geq 0} {\mathbb E}[ | w(t) |_{{\mathcal H}}^2 + m( | B^{1/2} u_0 |_H^2 )] < \infty, \tag{2} \] where \({\mathcal H} = D(A) \times H\), \(B\) is the Laplace operator with fixed boundary conditions, and \(m\) : \([0, \infty)\) \(\to\) \([0, \infty)\) is a \(C^1\)-class increasing function such that \(m(0) =0\) and \(\Phi(u)\) \(=\) \(\frac{1}{2} m( | B^{1/2} u |^2 )\) for \(u\) \(\in\) \(D(A)\).
For other related works, see, e.g., [the second author et al., Probab. Theory Relat. Fields 132, No. 1, 119–149 (2005; Zbl 1071.60053)] for stochastic nonlinear beam equations, [S. Albeverio et al., J. Math. Anal. Appl. 371, No. 1, 309–322 (2010; Zbl 1197.60050)] for invariant measures of SDEs driven by Poisson type noise, and [the second author et al., Nonlinear Anal., Real World Appl. 17, 283–310 (2014; Zbl 1310.60091)] for SPDEs driven by Lévy noise.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35L70 Second-order nonlinear hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60J75 Jump processes (MSC2010)
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[1] Aït-Sahalia, Telling from discrete data whether the underlying continuous-time model is a diffusion,, Journal of Finance, 57, 2075, (2002)
[2] S. Albeverio, 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, 309, (2010) · Zbl 1197.60050
[3] Z. Brzeźniak, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise,, Nonlinear Analysis: Real World Applications, 17, 283, (2014) · Zbl 1310.60091
[4] W. E. Baylis, Energy balance with the Landau-Lifshitz equation,, Phys. Lett. A, 301, 7, (2002) · Zbl 0997.70019
[5] D. Burgreen, <em> Free Vibrations of a Pin-Ended Column with Constant Distance Between Pin Ends,</em>, No. PIBAL-166. POLYTECHNIC INST OF BROOKLYN NY, (1950)
[6] Z. Brzeźniak, Stochastic differential equations on Banach manifolds,, Methods Funct. Anal. Topology, 6, 43, (2000) · Zbl 0965.58028
[7] Z. Brzeźniak, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces,, Stochastic Processes and Their Applications, 84, 187, (1999) · Zbl 0996.60074
[8] Z. Brzeźniak, Stochastic nonlinear beam equations,, Probability Theory and Related Fields 132 (2005), 132, 119, (2005) · Zbl 1071.60053
[9] Z. Brzeźniak, Martingale solutions for stochastic equation of reaction diffusion type driven by Lévy noise or Poisson random measure, preprint,, <a href=
[10] J. F. Burrow, Lévy processes, saltatory foraging, and superdiffusion,, Mathematical Modelling of Natural Phenomena 3 (2008), 3, 115, (2008) · Zbl 1337.92171
[11] A. Carroll, <em>The Stochastic Nonlinear Heat Equation</em>,, Ph. D. Thesis, (1999)
[12] P. L. Chow, Stochastic PDE for nonlinear vibration of elastic panels,, Differential Integral Equations, 12, 419, (1999) · Zbl 1006.60058
[13] S. R. Das, The surprise element: Jumps in interest rates,, Journal of Econometrics, 106, 27, (2002) · Zbl 1051.62106
[14] J. G. Eisley, Nonlinear vibration of beams and rectangular plates,, Zeitschrift für angewandte Mathematik und Physik ZAMP, 15, 167, (1964) · Zbl 0133.19101
[15] D. Gilbarg, <em>Elliptic partial differential equations of second order</em>,, Reprint of the 1998 edition, (1998)
[16] I. Gyöngy, On Stochastic Equations with Respect to Semimartingales. I,, Stochastics: An International Journal of Probability and Stochastic Processes, 4, 1, (1980) · Zbl 0439.60061
[17] I. Gyöngy, On stochastic equations with respect to semimartingale III,, Stochastics: An International Journal of Probability and Stochastic Processes, 7, 231, (1982) · Zbl 0495.60067
[18] E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure,, Electron. J. Probab, 10, 1496, (2005) · Zbl 1109.60048
[19] P. D. Lax, <em>Scattering Theory</em>,, Pure and Applied Mathematics, (1967) · Zbl 0214.12002
[20] R. Z. Khas’minskii, Stability of systems of differential equations under random perturbations of their parameters,, Izdat. , (1969) · Zbl 0214.15903
[21] B. Maslowski, Integral continuity and stability for stochastic hyperbolic equations,, Differential Integral Equations, 6, 355, (1993) · Zbl 0777.35096
[22] M. Métivier, <em>Semimartingales, A Course on Stochastic Processes</em>,, de Gruyter Studies in Mathematics, (1982) · Zbl 0503.60054
[23] M. Ondreját, a private communication to, [8] · Zbl 1238.60073
[24] S. K. Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation,, Journal of Differential Equations, 135, 299, (1997) · Zbl 0884.35105
[25] S. Peszat, <em>Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach</em>,, Encyclopedia of Mathematics and its Applications, (2007) · Zbl 1205.60122
[26] A. J. Pritchard, Stability and stabilizability of infinite-dimensional systems,, SIAM Review, 23, 25, (1981) · Zbl 0452.93029
[27] M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes,, Potential Analysis, 42, 809, (2015) · Zbl 1332.60026
[28] T. Russo, Lévy processes and stochastic von Bertalanffy models of growth, with application to fish population analysis,, Journal of Theoretical Biology, 258, 521, (2009)
[29] L. Tubaro, On abstract stochastic differential equation in Hilbert spaces with dissipative drift,, Stochastic Analysis and Applications, 1, 205, (1983) · Zbl 0509.60063
[30] L. Tubaro, An estimate of Burkholder type for stochastic processes defined by the stochastic integral,, Stochastic Analysis and Applications, 2, 187, (1984) · Zbl 0539.60056
[31] J. Van Neerven, A maximal inequality for stochastic convolutions in 2-smooth Banach spaces,, Electron. Commun. Probab, 16, 689, (2011) · Zbl 1254.60053
[32] H. Triebel, <em> Interpolation Theory, Function Spaces, Differential Operators</em>,, North-Holland Mathematical Library, (1978) · Zbl 0387.46033
[33] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17, 35, (1950) · Zbl 0036.13302
[34] J. Zhu, <em>A Study of SPDEs w.r.t. Compensated Poisson Random Measures and Related Topics</em>,, Ph. D. Thesis, (2010)
[35] J. Zhu, Maximal inequality of stochastic convolution driven by compensated Poisson random measures in Banach spaces, preprint,, <a href=
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