Quantitative estimates of discrete harmonic measures
| dc.creator | Bolthausen, E. | |
| dc.creator | Muench-Berndl, K. | |
| dc.date | 1999-08-11 | |
| dc.date | 2000-05-05 | |
| dc.date.accessioned | 2026-07-07T05:30:16Z | |
| dc.date.available | 2026-07-07T05:30:16Z | |
| dc.description | A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all $\b>0$ there exists $ρ(d,\b)<d$ and $N(d,\b)$, such that for any $n>N(d,\b)$, any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ $$ | \{y\in\Z^d\colon ν_{A,x}(y) \geq n^{-\b} \}| \leq n^{ρ(d,\b)}, $$ where $ν_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $ρ$ must converge to $d$ as $\b \to \infty$. | |
| dc.description | 16 pages, 2 figures. Part (B) of the theorem is new | |
| dc.identifier | https://arxiv.org/abs/math/9908047 | |
| dc.identifier | http://arxiv.org/abs/math/9908047 | |
| dc.identifier | Israel Journal of Mathematics 124, 125-141 (2001) | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78936 | |
| dc.subject | Probability | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Quantitative estimates of discrete harmonic measures | |
| dc.type | text |