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

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.

### 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
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