Brownian motion and harmonic measure in conic sections

AbstractWe prove some elementary intuitive estimates on moving boundaries hitting times by one-dimensional Brownian motion (in ℝ and on the circle). These results give an alternative approach to Beurling's radial projection theorem on harmonic measure in a disc.

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Harmonic measure on 3-dimensional Brownian paths

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065567 ◽

1988 ◽

Vol 104 (2) ◽

pp. 407-412 ◽

Cited By ~ 1

Author(s):

Krzysztof Burdzy

Keyword(s):

Brownian Motion ◽

Harmonic Measure ◽

Occupation Measure ◽

3 Dimensional ◽

Positive Capacity

AbstractIt is well known that the trace X([0, ∞) of the 3-dimensional Brownian motion X has positive capacity and, therefore, the harmonic measure is well defined on this set. It is shown that a.s., this harmonic measure is singular with respect to the occupation measure Lebesgue ο X−1.

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Brownian motion and sets of harmonic measure zero

Pacific Journal of Mathematics ◽

10.2140/pjm.1981.95.179 ◽

1981 ◽

Vol 95 (1) ◽

pp. 179-192 ◽

Cited By ~ 13

Author(s):

Bernt Oksendal

Keyword(s):

Brownian Motion ◽

Harmonic Measure ◽

Measure Zero

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Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200003041 ◽

2007 ◽

Vol 44 (02) ◽

pp. 393-408 ◽

Cited By ~ 1

Author(s):

Allan Sly

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

White Noise ◽

Gaussian Process ◽

Discrete Data ◽

Hölder Exponent ◽

Scaling Properties ◽

Multifractional Brownian Motion ◽

Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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Brownian motion with quadratic killing and some implications

Journal of Applied Probability ◽

10.1017/s0021900200116079 ◽

1986 ◽

Vol 23 (04) ◽

pp. 893-903 ◽

Cited By ~ 2

Author(s):

Michael L. Wenocur

Keyword(s):

Brownian Motion ◽

Weibull Distribution ◽

Special Function ◽

Function Theory ◽

Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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