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Title: Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures

Abstract

In this paper, we study a class of stochastic nonlinear wave equation with memory driven by Lévy noise. We first show the existence and uniqueness of global mild solutions using a suitable energy function. Second, under some additional assumptions we prove the exponential stability of the solutions.

Authors:
 [1];  [2];  [3]
  1. Department of Mathematics, Northwest University, Xi An 710069 (China)
  2. (China)
  3. Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Science, Nanjing Normal University, Nanjing 210023 (China)
Publication Date:
OSTI Identifier:
22250899
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 3; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FUNCTIONS; MATHEMATICAL SOLUTIONS; NOISE; NONLINEAR PROBLEMS; RANDOMNESS; STABILITY; STOCHASTIC PROCESSES; WAVE EQUATIONS

Citation Formats

Liang, Fei, Department of Mathematics, Xi An University of Science and Technology, Xi An 710054, and Gao, Hongjun, E-mail: gaohj@njnu.edu.cn. Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures. United States: N. p., 2014. Web. doi:10.1063/1.4867614.
Liang, Fei, Department of Mathematics, Xi An University of Science and Technology, Xi An 710054, & Gao, Hongjun, E-mail: gaohj@njnu.edu.cn. Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures. United States. doi:10.1063/1.4867614.
Liang, Fei, Department of Mathematics, Xi An University of Science and Technology, Xi An 710054, and Gao, Hongjun, E-mail: gaohj@njnu.edu.cn. 2014. "Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures". United States. doi:10.1063/1.4867614.
@article{osti_22250899,
title = {Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures},
author = {Liang, Fei and Department of Mathematics, Xi An University of Science and Technology, Xi An 710054 and Gao, Hongjun, E-mail: gaohj@njnu.edu.cn},
abstractNote = {In this paper, we study a class of stochastic nonlinear wave equation with memory driven by Lévy noise. We first show the existence and uniqueness of global mild solutions using a suitable energy function. Second, under some additional assumptions we prove the exponential stability of the solutions.},
doi = {10.1063/1.4867614},
journal = {Journal of Mathematical Physics},
number = 3,
volume = 55,
place = {United States},
year = 2014,
month = 3
}
  • In this paper, we study a class of stochastic nonlinear wave equation with memory driven by Lévy noise. We first show the existence and uniqueness of global mild solutions using a suitable energy function. Second, under some additional assumptions we prove the exponential stability of the solutions.
  • The thesis is devoted primarily to the study of stochastic differential equations on duals of nuclear spaces driven by Poisson random measures. The existence of a weak solution is obtained by the Galerkin method and the uniqueness is established by implementing the Yamada-Watanabe argument in the present setup. When the magnitudes of the driving terms are small enough and the Poisson streams occur frequently enough, it is proved that the stochastic differential equations mentioned above can be approximated by diffusion equations. Finally, the author considers a system of interacting stochastic differential equations driven by Poisson random measures. Let (X[sup n][submore » i](t), [center dot][center dot][center dot], X[sup n][sub n](t)) be the solution of this system and consider the empirical measures [zeta]n([omega],B) [identical to] (1/n) (sum of j=1 to n) [delta]x[sup n][sub j]([center dot],[omega])(B) (n[>=]1). It is provided that [zeta][sub n] converges in distribution to a non-random measure which is the unique solution of a McKean-Vlasov equation. The above problems are motivated by applications to neurophysiology, in particular, to the fluctuation of voltage potentials of spatially distributed neurons and to the study of asymptotic behavior of large systems of interacting neurons.« less
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