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Lyapunov exponents of hybrid stochastic heat equations. (English) Zbl 1250.93122
Summary: In this paper, we investigate a class of hybrid stochastic heat equations. By explicit formulae of solutions, we not only reveal the sample Lyapunov exponents but also discuss the \(p\)-th moment Lyapunov exponents. Moreover, several examples are established to demonstrate that unstable (deterministic or stochastic) dynamical systems can be stabilized by Markovian switching.

MSC:
93E15 Stochastic stability in control theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J75 Jump processes (MSC2010)
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[1] Arnold, L.; Crauel, H.; Wihstutz, V., Stabilization of linear systems by noise, SIAM J. control optim., 21, 451-461, (1983) · Zbl 0514.93069
[2] Has’minskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoff and Noord-hoff Alphen aan den Rijn, The Netherlands · Zbl 0276.60059
[3] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press · Zbl 1126.60002
[4] Pardoux, E.; Wihstutz, V., Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion, SIAM J. appl. math., 48, 442-457, (1988) · Zbl 0641.60065
[5] Pardoux, E.; Wihstutz, V., Lyapounov exponent of linear stochastic systems with large diffusion term, Stoch. proc. appl., 40, 289-308, (1992) · Zbl 0749.60061
[6] Scheutzow, M., Stabilization and destabilization by noise in the plane, Stoch. anal. appl., 11, 97-113, (1993) · Zbl 0766.60072
[7] Kwiecinska, A.A., Stabilization of partial differential equations by noise, Stoch. proc. appl., 79, 179-184, (1999) · Zbl 0962.60052
[8] Kwiecinska, A.A., Stabilization of evolution equations by noise, Proc. amer. math. soc., 130, 3067-3074, (2002) · Zbl 1003.35074
[9] Caraballo, T.; Liu, K.; Mao, X., On stabilization of partial differential equations by noise, Nagoya math. J., 161, 155-170, (2001) · Zbl 0986.60058
[10] Blömker, D.; Hairer, M.; Pavliotis, G.A., Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20, 1721-1744, (2007) · Zbl 1138.60004
[11] Yin, G.; Zhu, C., ()
[12] Kwiecinska, A.A., Almost sure and moment stability of stochastic partial differential equations, Probab. math. statist., 21, 405-415, (2001) · Zbl 1004.60070
[13] Xie, B., The moment and almost sure exponential stability of stochastic heat equation, Proc. amer. math. soc., 136, 3627-3634, (2008) · Zbl 1147.93395
[14] Donsker, M.D.; Varadhan, S.R.S., Asymptotic evaluation of certain Markov process expectations for large time, I-IV, Comm. pure appl. math., 28, 1-47, (1975), 279-301, 29 (1976), 389-461, 36 (1983), 183-212 · Zbl 0323.60069
[15] Wu, L., Uniformly integrable operators and large deviations for Markov processes, J. funct. anal., 172, 301-376, (2000) · Zbl 0957.60032
[16] Da Prato, G.; Iannelli, M.; Tubaro, L., Some results on linear stochastic differential equations in Hilbert spaces, Stochastics, 6, 105-116, (1982) · Zbl 0475.60041
[17] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052
[18] Gong, G.; Qian, M., On the large deviation functions of Markov chains, Acta math. sci., (English ed.), 8, 199-209, (1988) · Zbl 0682.60019
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