Central limit theorem for a random walk with random obstacles in \(R^ d\). (English) Zbl 0783.60108

Tanemura, Hideki

Ann. Probab. 21, No. 2, 936-960 (1993).

The process \((X_ t,\mathbb{P}_ \eta)\) considered in the paper consists of two parts: First, for each \(\eta\subset \mathbb{R}^ d\), \(\eta\) being finite on compacts, \(X_ t\) is an \(\mathbb{R}^ d\)-valued jump process with generator \[ L_ \eta \varphi(x)=\int_{\mathbb{R}^ d}dy(\varphi(y)- \varphi(x))p(| x-y| )\exp\left[-\sum_{u\in \eta,\;v=x,y}\Psi(| u-v| )\right], \] where \(p(\cdot): \mathbb{R}_ +\to \mathbb{R}_ +\), \(\int_{\mathbb{R}^ d}p(| x|)dx=1\) and \(\Psi: \mathbb{R}_ +\to \mathbb{R}^ d\) is bounded below and \(\Psi=\infty\) on \([0,r)\), \(=0\) on \((r_ 0,\infty)\) for some constants \(0<r\leq r_ 0\). Second, the environment process \((\eta_ t)\) is a reversible Markov process with a pair potential and with an ergodic Gibbsian distribution \(\mu^*\) having an unbounded support. Under some restriction on \(\mu^*\), the author proves a central limit theorem. That is, under \(\mathbb{P}_{\mu^*}\), \(\varepsilon X_{t/\varepsilon^ 2}\) converges as \(\varepsilon\to 0\) in distribution to a scaled Brownian motion. A similar result for a tagged particle is also obtained. It is interesting that some continuum analogs of the site percolation and electrical network are adopted to derive the conclusion.

Reviewer: Chen Mu-fa (Beijing)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G50	Sums of independent random variables; random walks
60F05	Central limit and other weak theorems

Keywords:

Gibbsian distribution; central limit theorem; Brownian motion; percolation; electrical network