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In this paper we consider the simple random walk on $\mathbb{Z}^d$, $d \geq 3$, conditioned to stay forever in a large domain $D_N$ of typical diameter $N$. Considering the range up to time $t_N \geq N^{2+\delta}$ for so...
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We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set Ω ⊂ R d , in dimension d ≥ 2. More precisely, consider the simple random walk on Z d killed upon exiting the (...