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Escape rate for $ 2$-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 $ P\{ X(t) < f(t)$ i.o. as $ t \uparrow \infty \} = 0$ or 1 according as $ \sum\nolimits_{n = 1}^\infty {{e^{ - n}}\log f({e^{{e^n}}}) < } $ or $ = \infty $. The result is used to construct a complex-valued Zygmund function--as a lacunary series--whose graph does not have $ \sigma $-finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have $ \sigma $-finite linear Hausdorff measure.


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Additional Information

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

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