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# Escape Rate for 2-Dimensional Brownian Motion Conditioned to be Transient and Application to Zygmund Functions

Elizabeth Ann Housworth
Transactions of the American Mathematical Society
Vol. 343, No. 2 (Jun., 1994), pp. 843-852
DOI: 10.2307/2154745
Stable URL: http://www.jstor.org/stable/2154745
Page Count: 10
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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 $\sum^\infty_{n = 1} e^{-n} \log f(e^{e^n}) < \text{or} = \infty$. The result is used to construct a complex-valued Zygmund function--as a lacunary series--whose graph does not have σ-finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have σ-finite linear Hausdorff measure.

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