Brownian motion and sets of harmonic measure zero

Øksendal, Bernt

doi:10.2140/pjm.1981.95.179

Cited by 26 publications

(14 citation statements)

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“…In his remarkable paper [Mkl], N. G. Makarov proved that (1) oj lAo for all a > 1, solving the conjecture due to Oksendal [0]. In [Mk2], Makarov gave a proof of the equality HD(a>) = 1 for every Jordan domain (this result was generalized afterwards for an arbitrary simply connected domain).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Harmonic Measure Versus Hausdorff Measures on Repellers for Holomorphic Maps

Zdunik¹

1991

Transactions of the American Mathematical Society

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Abstract. This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbañski. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the Hausdorff measure corresponding to the function c(t) = íexp(c\/log | log log log j) for some c > 0 .

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Harmonic Measure Versus Hausdorff Measures on Repellers for Holomorphic Maps

Zdunik¹

1991

Transactions of the American Mathematical Society

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“…In [7], Θksendal proved that if D is a simply-connected domain in R 2 , and E is a set on dD with vanishing linear measure, and if E is situated on a line, then E has vanishing harmonic measure ω(E, D) with respect to D. In [3], Kaufman and Wu generalized this result and proved that the theorem still holds if E is situated on a quasi-smooth curve, but no longer holds if E is situated on a quasi-conformal circle. An interesting, perhaps very difficult, question is whether the theorem is true if E lies on a rectifiable curve.…”

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confidence: 99%

On singularity of harmonic measure in space

Wu¹

1986

Pacific J. Math.

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“…It is also closely related to Theorem 2 in [8] and Theorem 3 in [7] when m = 2 and D is simply-connected. However, the present theorem does not imply the results in [7 or 8], because, for a set F lying on the boundary of a simply-connected domain, we may not be able to find T so that the conditions (1) and (2) are both fulfilled.…”

Section: Examples On Harmonic Measure and Normal Numbers1mentioning

confidence: 85%

“…The example is particularly interesting when ai -0 (or a2 = 0). Some complementary examples on domains whose boundaries consist of other Cantor sets can be found in [8].…”

Section: Examples On Harmonic Measure and Normal Numbers1mentioning

confidence: 99%

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Examples on harmonic measure and normal numbers

Wu¹

1985

Proc. Amer. Math. Soc.

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ABSTRACT. Suppose that F is a bounded set in Rm, m > 2, with positive capacity. Add to F a disjoint set E so that E U F is closed, and let D = Rm\ (ËUF).Under what conditions on the added set E do we have harmonic measure u(F, D) = 0? It turns out that besides the size of E near F, the location of E relative to F also plays an important role. Our example, based on normal numbers, stresses this fact.Suppose that F is a bounded set in Rm, m > 2, with positive capacity. Add to F a disjoint set E so that E U F is closed, and let D = Rm\(E U F). Under what conditions on the added set E do we have harmonic measure w(F, D) = 0? It turns out that besides the size of E near F, the location of E relative to F also plays an important role. Our example, based on normal numbers, stresses this fact.THEOREM. Let D be a bounded domain in Rm, m > 2, F be a subset of dD with Am~1(F) = 0, and E = Rm\(D U F). Suppose that F lies also on some quasi-smooth curve Y when m = 2, on some BMOi surface T when m > 3. And suppose that at each a G F, 0 < r < |, there is a closed set T Ç EC\B(a, r) so that (1) capacity ( However, when m = 2, (1) is more restrictive than (1'). The surface T and dD are in general distinct; no smoothness condition is imposed on dD. The problem is very different if dD is quasi-smooth or BMOi (see [4]). When m > 3, topological properties of D are less important in studying u(F,D):

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Brownian motion and sets of harmonic measure zero

Cited by 26 publications

References 10 publications

Harmonic Measure Versus Hausdorff Measures on Repellers for Holomorphic Maps

Harmonic Measure Versus Hausdorff Measures on Repellers for Holomorphic Maps

On singularity of harmonic measure in space

Examples on harmonic measure and normal numbers

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