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Homogenization of a bond diffusion in a locally ergodic random environment. (English) Zbl 1075.60129
A nearest neighbour random walk on $$\mathbb Z$$ with random jump rates is studied. It is supposed that the jump rate from state $$x$$ to state $$x+1$$ is a strictly positive bounded random variable $$\eta (x)$$, $$0<c_ {-}\leq \eta (x)\leq c_ {+}<\infty$$, and $$\eta (x)$$ equals also to the jump rate from $$x+1$$ to $$x$$. Let $$\Omega = [c_ {-},c_ {+}] ^ {\mathbb Z}$$ be the space of configurations, and $$(\tau _ {z},\; z\in \mathbb Z)$$ the group of translations on $$\Omega$$, $$\tau _ {z}\eta (x) = \eta (x+z)$$. For a fixed environment $$\eta \in \Omega$$, let $$(x^ \eta _ {t})_ {t\geq 0}$$ be the corresponding random walk with $$x^ \eta _ 0 = 0$$; denote by $$X^ {\eta ,\varepsilon }$$ a rescaled process defined by $$X^ {\eta , \varepsilon }_ {t} = \varepsilon x^ \eta _ {\varepsilon ^ {-2}t}$$. Further it is assumed that $$(\mu _ \varepsilon )_ {\varepsilon >0}$$ is a locally ergodic system of probability measures on $$\Omega$$, that is, there exists a family $$(\bar \mu _ {y},\; y\in \mathbb R)$$ of probability measures which are ergodic with respect to the translation group $$(\tau _ {z})$$ and such that $\varepsilon \sum _ {z} g(\varepsilon z)f(\varepsilon z,\tau _ {z} \eta ) \longrightarrow \iint g(y)f(y,\omega )\,d\bar \mu _ {y} (\omega )\,dy \quad \text{as $$\varepsilon \downarrow 0$$}$ in $$\mu _ \varepsilon$$-probability for any bounded measurable function $$f:\mathbb R\times \Omega \to \mathbb R$$ continuous in the first variable and local in the second one, and for any compactly supported continuous function $$g:\mathbb R\to \mathbb R$$. Finally, it is supposed that the function $$y\mapsto \int f\,d\bar \mu _ {y}$$ is locally integrable on $$\mathbb R$$ for any bounded local function $$f$$ on $$\Omega$$, and the function $$a: y\mapsto (\int \eta (0)^ {-1} \,d\bar \mu _ {y})^ {-1}$$ satisfies $$a\in C^ 2(\mathbb R)$$ and has bounded derivatives. Under these hypotheses it is proven that the distribution of $$X^ {\eta ,\varepsilon }$$ converges in $$\mu _ \varepsilon$$-probability to the law of a diffusion process on $$\mathbb R$$ with the generator $$Lf = {d\over dy}(a(y){df\over dy})$$, $$f\in C^ 2(\mathbb R)$$.

##### MSC:
 60K37 Processes in random environments 60F17 Functional limit theorems; invariance principles 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
##### Keywords:
random walk in random environment
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##### References:
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