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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.

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
Full Text: DOI EuDML
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