## Escape rate for $2$-dimensional Brownian motion conditioned to be transient with application to Zygmund functions

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- by Elizabeth Ann Housworth
- Trans. Amer. Math. Soc.
**343**(1994), 843-852 - DOI: https://doi.org/10.1090/S0002-9947-1994-1222193-0
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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.## References

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## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- 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