×

zbMATH — the first resource for mathematics

On a stochastic wave equation driven by a non-Gaussian Lévy process. (English) Zbl 1196.60113
The authors study a hyperbolic equation with a non-Gaussian noise perturbation. The existence, uniqueness and asymptotic behaviour of the weak solutions to white-noise perturbed stochastic wave equations have been investigated by several authors. Specially, Peszat and Zabczyk considered the following wave equation driven by an impulsive noise: \[ \frac{\partial^2u(t)}{\partial t^2}= [\Delta u(t) + f(u(t))]dt + b(u(t))PdZ(t) \] where \(f, b\) are Lipschitz continuous real functions, \(P\) is a regularizing linear operator, and the impulsive noise \(Z_t\) is formulated as a Poisson random measure. Comparing with this work, the authors’ objective is first to consider the more general damped wave equation which is used to model nonlinear phenomena in relativistic quantum mechanics. Their focus is on the notion of weak solution which is stronger than the mild solution. Moreover, the perturbation can inculde a general non-Gaussian Lévy noise. The existence of a unique invariant measure is also explored.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K90 Abstract parabolic equations
47D07 Markov semigroups and applications to diffusion processes
35R30 Inverse problems for PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Barbu, V., Da Prato, G., Tubaro, L.: Stochastic wave equations with dissipative damping. Stoch. Process. Appl. 117, 1001–1013 (2007) · Zbl 1122.60056 · doi:10.1016/j.spa.2006.11.006
[2] Bo, L., Shi, K., Wang, Y.: Stochastic wave equation driven by compensated Poisson random measure. Preprint (2008)
[3] Bo, L., Tang, D., Wang, Y.: Explosive solutions of stochastic wave equations with damping on R d . J. Differ. Equ. 244, 170–187 (2008) · Zbl 1129.60056 · doi:10.1016/j.jde.2007.10.016
[4] Brzeniak, Z., Maslowski, B., Seidler, J.: Stochastic nonlinear beam equations. Probab. Theory Relat. Fields 132, 119–149 (2005) · Zbl 1071.60053 · doi:10.1007/s00440-004-0392-5
[5] Chow, P.: Stochastic wave equation with polynomial nonlinearity. Ann. Appl. Probab. 12, 361–381 (2002) · Zbl 1017.60071 · doi:10.1214/aoap/1015961168
[6] Chow, P.: Asymptotics of solutions to semilinear stochastic wave equations. Ann. Appl. Probab. 16, 757–789 (2006) · Zbl 1126.60051 · doi:10.1214/105051606000000141
[7] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981) · Zbl 0495.60005
[8] Lions, J.L.: Equations différentielles operationelles et problèmes aux limits. Springer, Berlin (1961)
[9] Peszat, S., Zabczyk, J.: Stochastic heat and wave equations driven by an impulsive noise. In: Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications, VII. Lect. Notes Pure Appl., vol. 245, pp. 229–242. Chapman & Hall/CRC, Boca Raton (2006) · Zbl 1115.60320
[10] Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy noise: An Evolution Equation Approach. Encyclopedia of Mathematics and Its Applications, vol. 113. Cambridge University Press, Cambridge (2007) · Zbl 1205.60122
[11] Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) · Zbl 0973.60001
[12] Schiff, L.: Nonlinear meson theory of nuclear forces, I. Phys. Rev. 84, 1–9 (1951) · Zbl 0054.08305 · doi:10.1103/PhysRev.84.1
[13] Segal, I.: The global Cauchy problem for a relativistic scalar field with power interaction. Bull. Soc. Math. Fr. 91, 129–135 (1963) · Zbl 0178.45403
[14] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997) · Zbl 0871.35001
[15] Zeidler, E.: Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators. Springer, New York (1990) · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.