×

zbMATH — the first resource for mathematics

Lyapunov exponents of PDEs driven by fractional noise with Markovian switching. (English) Zbl 1336.60119
Summary: In this article, we study a class of stochastic parabolic equations driven by fractional noise with Markovian switching. Based on the explicit representation of the strong solution given by an evolution system, we investigate the \(p\)-th moment and almost sure exponential stabilities with the exponential rate function \(t^{2H}\).

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bao, J.; Mao, X.; Yuan, C., Lyapunov exponents of hybrid stochastic heat equations, Systems Control Lett., 61, 165-172, (2012) · Zbl 1250.93122
[2] Biagini, F.; Hu, Y.; Öksendal, B.; Zhang, T., Stochastic calculus for fractional Brownian motion and applications, (2008), Springer-Verlag London
[3] Bo, L.; Jiang, Y.; Wang, Y., On a class of stochastic Anderson models with fractional noises, Stoch. Anal. Appl., 26, 256-273, (2008) · Zbl 1136.60345
[4] Donsker, M. D.; Varadhan, S. R.S., Asymptotic evaluation of certain Markov process expectations for large time, I, Comm. Pure Appl. Math., 28, 1-47, (1975) · Zbl 0323.60069
[5] Duncan, T. E.; Maslowski, B.; Pasik-Duncan, B., Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115, 1357-1383, (2005) · Zbl 1076.60054
[6] Hu, Y., Heat equations with fractional white noise potentials, Appl. Math. Optim., 43, 221-243, (2001) · Zbl 0993.60065
[7] Hunt, G. A., Random Fourier transforms, Trans. Amer. Math. Soc., 71, 38-69, (1951) · Zbl 0043.30601
[8] Jiang, Y.; Shi, K.; Wang, Y., Stochastic fractional Anderson models with fractional noises, Chinese Ann. Math., 31B, 101-118, (2010) · Zbl 1185.35346
[9] Kwiecińska, A. A., Stabilization of partial differential equations by noise, Stochastic Process. Appl., 79, 179-184, (1999) · Zbl 0962.60052
[10] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press · Zbl 1126.60002
[11] Mémin, J.; Mishura, Y.; Valkeila, E., Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett., 51, 197-206, (2001) · Zbl 0983.60052
[12] Móricz, F. A.; Serfling, R. J.; Stout, W. F., Moment and probability bounds with quasi-superadditive structure for the maximum partial sum, Ann. Probab., 10, 1032-1040, (1982) · Zbl 0499.60052
[13] Nane, E.; Xiao, Y.; Zeleke, A., A strong law of large numbers with applications to self-similar stable processes, Acta Sci. Math. (Szeged), 76, 697-711, (2010) · Zbl 1274.60098
[14] Nualart, D., (The Malliavin Calculus and Related Topics, Probability and Its Applications, (New York), (2006), Springer Berlin)
[15] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[16] Tanabe, H., Equations of evolutions, (1979), Pitman London
[17] Xi, F.; Yin, G., Jump-diffusions with state-dependent switching: existence and uniqueness, Feller property, linearization, and uniform ergodicity, Sci. China-Math., 54, 2651-2667, (2011) · Zbl 1274.60253
[18] Xie, B., The moment and almost sure exponential stability of stochastic heat equation, Proc. Amer. Math. Soc., 136, 3627-3634, (2008) · Zbl 1147.93395
[19] Yin, G.; Zhu, C., Hybrid switching diffusions: properties and applications, Stochastic Modelling and Applied Probability, vol. 63, (2010), Springer New York · Zbl 1279.60007
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.