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Homogenization of random semilinear PDEs. (English) Zbl 0989.35022
A homogenization problem for a system of semilinear random parabolic equations $\partial_{t}u^\varepsilon + \frac 12 e^{2V(x/\varepsilon,\omega)} \text{div}\left(a\Bigl({x\over\varepsilon},\omega\Bigr)e^{-2 V(x/\varepsilon,\omega)}\nabla u^\varepsilon\right) + h(x, u^\varepsilon,\nabla u^\varepsilon) = 0, \;u^\varepsilon(T, \cdot) = H,$ with $$u=(u_1,\ldots,u_{m})$$, $$x\in\mathbb R^{d}$$, and $$t\in [0,T]$$, is concerned. Let $$(\Omega,\mu)$$ be a probability space, $$(\tau_{x}, x\in\mathbb R^{d})$$ be a group of transformations of $$\Omega$$ satisfying measurability and stochastic continuity assumptions and such that $$\mu$$ is invariant and ergodic under the action of $$(\tau_{x})$$. Let $$\mathbf V:\Omega\to\mathbb R$$ be an essentially bounded measurable function, $$\mathbf a:\Omega\to\mathbb R^{d}\otimes \mathbb R^{d}$$ a measurable function whose values are symmetric matrices; the random fields $$a$$, $$V$$ are defined by $$a(x,\omega) = \mathbf a(\tau _{x}\omega)$$ and $$V(x,\omega) = \mathbf V(\tau_{x}\omega)$$. Assume that $$\mathbf a$$ is positive definite and bounded $$\mu$$-almost surely and $$\langle \mu, e^{-2\mathbf V}\rangle = 1$$. Let the function $$H:\mathbb R^{d}\to\mathbb R^{m}$$ be continuous and square integrable, let $$h:\mathbb R^{d}\times\mathbb R^{m}\times\mathbb R^{m \times d}\to\mathbb R^{m}$$ be a bounded uniformly continuous function satisfying some Lipschitzianity and monotonicity hypotheses. Then $$\lim_{\varepsilon\to 0}\int_\Omega\int_{G}\|u^\varepsilon(t,x)-u^0(t,x)\|^{p} dx d\mu =0$$ for all bounded domains $$G\subset \mathbb R^{d}$$, all $$t\in[0,T]$$ and all $$p\in\mathopen]1,Q\mathclose[$$ with certain $$Q\in\mathopen ]1,2\mathclose]$$. Here $$u^0$$ denotes the solution to the homogenized system $$\partial_{t}u^0 + \frac 12\overline a_{ij}\partial^2_{ij} u^0 + \overline h(x,u^0,\nabla u^0)=0$$, $$u^0(T,\cdot) = H$$. The coefficients $$\overline a$$, $$\overline h$$ are defined by $$\overline a = \int_\Omega (\text{Id}+\phi)\mathbf a(\text{Id}+\phi)^* e^{-2\mathbf V} d\mu$$, $$\overline h(x,y,z)= \int_\Omega h(x,y,z(\text{Id}+\phi))e^{ -2\mathbf V} d\mu$$, where $$\phi = (\phi^{i}_{j})$$ and $$\phi^{i}$$ are vortex-free stationary fields solving a suitable auxiliary problem.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35R60 PDEs with randomness, stochastic partial differential equations 60K37 Processes in random environments
##### Keywords:
random media; random parabolic equations
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