## Scaling limit of the prudent walk.(English)Zbl 1201.60029

Summary: We describe the scaling limit of the nearest neighbour prudent walk on $$Z^{2}$$, which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process
$Z_u =\int_0^{3u/7}(\sigma_1 \mathbf 1_{\{W(s)\geq 0\}} \vec e_1+\sigma_2\mathbf 1_{\{W(s)< 0\}} \vec e_2)\, ds,\quad u \in [0,1],$ where $$W$$ is the one-dimensional Brownian motion and $$\sigma_1,\sigma_2$$ two random signs. In particular, the asymptotic speed of the walk is well-defined in the $$L^1$$-norm and equals 3/7.

### MSC:

 60F17 Functional limit theorems; invariance principles 60G50 Sums of independent random variables; random walks 60G52 Stable stochastic processes
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