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Average and deviation for slow-fast stochastic partial differential equations. (English) Zbl 1251.35201
Summary: Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the stochastic deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of order \(\mathcal O(\epsilon )\) instead of order \(\mathcal O(\sqrt \epsilon)\) attained in previous averaging.

MSC:
35R60 PDEs with randomness, stochastic partial differential equations
60G44 Martingales with continuous parameter
60B10 Convergence of probability measures
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[1] Arnold, V.I.; Kozlov, V.V.; Neishtadt, A.I., Mathematical aspects of classical and celestial mechanics, () · Zbl 0885.70001
[2] Bogolyubov, N.N.; Mitropolskii, Y.A., Asymptotic methods in the theory of nonlinear oscillations, (1961), Hindustan Publ. Co.
[3] Cerrai, S., Second order PDEs in finite and infinite dimension. A probabilistic approach, Lecture notes in math., vol. 1762, (2001), Springer Heidelberg
[4] Cerrai, S.; Freidlin, M., Averaging principle for a class of stochastic reaction-diffusion equations, Probab. theory related fields, 144, 1-2, 137-177, (2009) · Zbl 1176.60049
[5] E, W.; Li, X.; Vanden-Eijnden, E., Some recent progress in multiscale modeling, (), 3-21 · Zbl 1419.74252
[6] Ethier, S.N.; Kurtz, T.G., Markov processes: characterization and convergence, (1986), John Wiley & Sons · Zbl 0592.60049
[7] Freidlin, M.I.; Wentzell, A.D., Random perturbations of dynamical systems, (1998), Springer Heidelberg · Zbl 0922.60006
[8] Kesten, H.; Papanicolaou, G.C., A limit theorem for turbulent diffusion, Comm. math. phys., 65, 79-128, (1979) · Zbl 0399.60049
[9] Khasminskii, R.Z., On the principle of averaging the itoʼs stochastic differential equations, Kibernetika, 4, 260-279, (1968), (in Russian)
[10] Kifer, Y., Diffusion approximation for slow motion in fully coupled averaging, Probab. theory related fields, 129, 157-181, (2004) · Zbl 1069.34070
[11] Imkeller, P.; Monahan, A., Stochastic climate dynamics, Stoch. dyn., 2, 3, (2002), (Special Issue)
[12] Metivier, M., Stochastic partial differential equations in infinite dimensional spaces, (1988), Scuola Normale Superiore Pisa · Zbl 0664.60062
[13] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052
[14] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. mat. pura appl., 146, 65-96, (1987) · Zbl 0629.46031
[15] Strook, D.W.; Varadhan, S.R.S., Multidimensional diffusion process, (1979), Springer Berlin-Heidelberg-New York
[16] Temam, R.; Miranville, A., Mathematical modeling in continuum mechanics, (2005), Cambridge University Press Cambridge
[17] Volosov, V.M., Averaging in systems of ordinary differential equations, Russian math. surveys, 17, 1-126, (1962) · Zbl 0119.07502
[18] Watanabe, H., Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients, Probab. theory related fields, 77, 359-378, (1988) · Zbl 0682.60047
[19] ()
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