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Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix. (English) Zbl 1207.60029
The paper deals with homogenization of second order PDEs with locally stationary coefficients. Namely, it is considered the strong solution \(X^\varepsilon\) of the following stochastic differential equation \[ X^\varepsilon_t= x+{1\over\varepsilon} \int^t_0 b\Biggl({X^\varepsilon_r\over\varepsilon}, X^\varepsilon_r\Biggr)\,dr+ \int^t_0 c\Biggl({X^\varepsilon_r\over \varepsilon}, X^\varepsilon_r\Biggr)\,dr+ \int^t_0\sigma \Biggl({X^\varepsilon_r\over\varepsilon}, X^\varepsilon_r\Biggr)\, dB_r, \] where for each \(y\in\mathbb{R}^d\) coefficients \(b(\cdot,y)\), \(c(\cdot,y)\), \(\sigma(\cdot,y)\) are stationary random fields independent of the \(d\)-dimensional standard Brownian motion \(B\). Thus \(b\), \(c\) and \(\sigma\) take into account both microscopic and macroscopic evolution scales. It is proved that, under suitable assumptions, the process \(X^\varepsilon\) converges weakly as \(\varepsilon\to 0\) in \(C([0, T];\mathbb{R}^d)\) to the solution of a stochastic differential equation with deterministic coefficients. The paper follows and improves the author’s earlier paper [Probab. Theor. Related Fields, 143, 545–568 (2009; Zbl 1163.60049)] by considering possibly degenerate diffusion matrices.

MSC:
60F17 Functional limit theorems; invariance principles
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35R60 PDEs with randomness, stochastic partial differential equations
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60J60 Diffusion processes
60K37 Processes in random environments
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References:
[1] A. Benchérif-Madani and E. Pardoux. Homogenization of a diffusion with locally periodic coefficients. In Séminaire de Probabilités XXXVIII 363-392. Lecture Notes in Math. 1857 . Springer, Berlin, 2005. · Zbl 1067.35009
[2] A. Bensoussan, J. L. Lions and G. Papanicolaou. Asymptotic Methods in Periodic Media . North Holland, Amsterdam, 1978. · Zbl 0404.35001
[3] F. Delarue and R. Rhodes. Stochastic homogenization of quasilinear PDEs with a spatial degeneracy. Asymptot. Anal. 61 (2009) 61-90. · Zbl 1180.35591 · doi:10.3233/ASY-2008-0925
[4] M. Fukushima. Dirichlet Forms and Markov Processes . North-Holland, Amsterdam, 1980. · Zbl 0422.31007
[5] M. Hairer and E. Pardoux. Homogenization of periodic linear degenerate PDEs. J. Funct. Anal. 255 2462-2487. · Zbl 1167.60015 · doi:10.1016/j.jfa.2008.04.014
[6] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaft 288 . Springer, Berlin, 1987. · Zbl 0635.60021
[7] V. V. Jikov, S. M. Kozlov and O. A. Oleinik. Homogenization of Differential Operators and Integral Functionals . Springer, Berlin, 1994.
[8] N. V. Krylov. Controlled Diffusion Processes . Springer, New York, 1980. · Zbl 0436.93055
[9] Z. M. Ma and M. Röckner. Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext . Springer, Berlin, 1992. · Zbl 0852.16022
[10] S. Olla. Homogenization of diffusion processes in Random Fields. Cours de l’école doctorale, Ecole polytechnique, 1994. Available at http://www.ceremade.dauphine.fr/ olla/pubolla.html.
[11] S. Olla and P. Siri. Homogenization of a bond diffusion in a locally ergodic random environment. Stochastic Process. Appl. 109 (2004) 317-326. · Zbl 1075.60129 · doi:10.1016/j.spa.2003.10.009
[12] R. Rhodes. On homogenization of space time dependent random flows. Stochastic Process. Appl. 117 (2007) 1561-1585. · Zbl 1127.60027 · doi:10.1016/j.spa.2007.01.010
[13] R. Rhodes. Diffusion in a locally stationary random environment. Probab. Theory Related Fields 143 (2009) 545-568. · Zbl 1163.60049 · doi:10.1007/s00440-007-0135-5
[14] D. Stroock. Diffusion semi-groups corresponding to uniformly elliptic divergence form operators. In Séminaires de Probabilités XXII 316-347. Lecture Notes in Math. 1321 . Springer, Berlin, 1988. (Section B 35 (1999) 121-141.) · Zbl 0651.47031 · numdam:SPS_1988__22__316_0 · eudml:113641
[15] L. Wu. Forward-Backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999) 121-141. · Zbl 0936.60037 · doi:10.1016/S0246-0203(99)80008-9 · numdam:AIHPB_1999__35_2_121_0 · eudml:77625
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