an:02233820
Zbl 1075.60129
Olla, S.; Siri, P.
Homogenization of a bond diffusion in a locally ergodic random environment
EN
Stochastic Processes Appl. 109, No. 2, 317-326 (2004).
00121374
2004
j
60K37 60F17 82D30
random walk in random environment
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 \(Lf = {d\over dy}(a(y){df\over dy})\), \(f\in C^ 2(\mathbb R)\).
Jan Seidler (Praha)