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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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