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

Abstract

The escape rate of a 2-dimensional Brownian motion conditioned to be transient is determined to be P { X ( t ) > f ( t ) P\{ X(t) > f(t) i.o. as t ↑ ∞ } = 0 t \uparrow \infty \} = 0 or 1 according as ∑ n = 1 ∞ e − n log ⁡ f ( e e n ) > \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 σ \sigma -finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have σ \sigma -finite linear Hausdorff measure.