Large deviations for stochastic heat equations with memory driven by Lévy-type noise

Markus Riedle; Jianliang Zhai

doi:10.3934/dcds.2018080

Article Contents

2018, Volume 38, Issue 4: 1983-2005. Doi: 10.3934/dcds.2018080

This issue Previous Article The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos Next Article Rarefaction waves for the Toda equation via nonlinear steepest descent

Large deviations for stochastic heat equations with memory driven by Lévy-type noise

Markus Riedle^1, and
Jianliang Zhai^2, ,

1.
Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom
2.
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

* Corresponding author
* Corresponding author

Received: January 2017

Revised: October 2017

Published: July 2018

The second author acknowledges funding by K. C. Wong Foundation for a 1-year fellowship at King's College London

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a Lévy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.

Keywords:

Mathematics Subject Classification: Primary: 60H15, 60F10; Secondary: 37L55.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, New York, NY: Academic Press, 2003.
	G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, New York: Springer, 2012.
	J. Bao and C. Yuan , Large deviations for neutral functional SDEs with jumps, Stochastics, 87 (2015) , 48-70. doi: 10.1080/17442508.2014.914516.
	V. Barbu , S. Bonaccorsi and L. Tubaro , Existence and asymptotic behavior for hereditary stochastic evolution equations, Appl. Math. Optim., 69 (2014) , 273-314. doi: 10.1007/s00245-013-9224-2.
	S. Bonaccorsi , G. da Prato and L. Tubaro , Asymptotic behavior of a class of nonlinear stochastic heat equations with memory effects, SIAM J. Math. Anal., 44 (2012) , 1562-1587. doi: 10.1137/110841795.
	A. Budhiraja , J. Chen and P. Dupuis , Large deviations for stochastic partial differential equations driven by a Poisson random measure, Stochastic Processes Appl., 123 (2013) , 523-560. doi: 10.1016/j.spa.2012.09.010.
	A. Budhiraja , P. Dupuis and V. Maroulas , Variational representations for continuous time processes, Ann. Inst. Henri Poincaré, Probab. Stat., 47 (2011) , 725-747. doi: 10.1214/10-AIHP382.
	T. Caraballo , I. D. Chueshov , P. Marín-Rubio and J. Real , Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007) , 253-270. doi: 10.3934/dcds.2007.18.253.
	T. Caraballo , J. Real and I. D. Chueshov , Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst., Ser. B, 9 (2008) , 525-539. doi: 10.3934/dcdsb.2008.9.525.
	B. D. Coleman and M. E. Gurtin , Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967) , 199-208. doi: 10.1007/BF01596912.
	Z. Dong , J. Xiong , J. Zhai and T. Zhang , A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017) , 227-254. doi: 10.1016/j.jfa.2016.10.012.
	C. Giorgi and V. Pata , Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, NoDEA, Nonlinear Differ. Equ. Appl., 8 (2001) , 157-171. doi: 10.1007/PL00001443.
	C. Giorgi , V. Pata and A. Marzocchi , Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA, Nonlinear Differ. Equ. Appl., 5 (1998) , 333-354. doi: 10.1007/s000300050049.
	B. Goldys , M. Röckner and X. Zhang , Martingale solutions and Markov selections for stochastic partial differential equations, Stochastic Processes Appl., 119 (2009) , 1725-1764. doi: 10.1016/j.spa.2008.08.009.
	M. E. Gurtin and B. C. Pipkin , A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968) , 113-126. doi: 10.1007/BF00281373.
	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Company, 1981.
	Y. Li , Y. Xie and X. Zhang , Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst., 35 (2015) , 5221-5237. doi: 10.3934/dcds.2015.35.5221.
	J. W. Nunziato , On heat conduction in materials with memory, Q. Appl. Math., 29 (1971) , 187-204. doi: 10.1090/qam/295683.
	J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by levy noise, 2017. Available from: https://www.e-publications.org/ims/submission/BEJ/user/submissionFile/27769?confirm=1f6bdfb2.
	X. Yang, J. Zhai and T. Zhang, Large deviations for SPDEs of jump type, Stoch. Dyn. , 15 (2015), 1550026, 30 pp.
	J. Zhai and T. Zhang , Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, Bernoulli, 21 (2015) , 2351-2392. doi: 10.3150/14-BEJ647.