Escape rate for -dimensional Brownian motion conditioned to be transient with application to Zygmund functions

Author:
Elizabeth Ann Housworth

Journal:
Trans. Amer. Math. Soc. **343** (1994), 843-852

MSC:
Primary 60J65; Secondary 30D40

DOI:
https://doi.org/10.1090/S0002-9947-1994-1222193-0

MathSciNet review:
1222193

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Abstract: The escape rate of a 2-dimensional Brownian motion conditioned to be transient is determined to be i.o. as or 1 according as or . The result is used to construct a complex-valued Zygmund function--as a lacunary series--whose graph does not have -finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have -finite linear Hausdorff measure.

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DOI:
https://doi.org/10.1090/S0002-9947-1994-1222193-0

Keywords:
Brownian motion,
killed processes,
generators,
scale functions,
Zygmund functions,
lacunary series

Article copyright:
© Copyright 1994
American Mathematical Society