BURKHOLDER’S EXIT-TIME CONDITION VIA PROPER MAPS: SMIRNOV REGULARITY, HARDY–ORLICZ CRITERIA AND TORSION-TYPE PROBLEMS
Résumé
Abstract Let $(Z_{t})_{t \geq 0}$ be a planar Brownian motion running in some domain W and denote by $\tau _{W}$ the exit time of $Z_{t}$ from W . To establish the finiteness of $\mathbf {E}(\sup _{0\leq t\leq \tau _{W}}|Z_{t}|^{p})$ from the finiteness of $\mathbf {E}(|Z_{\tau _{W}}|^{p})$ for some $p>0$ , Burkholder [‘Exit times of Brownian motion, harmonic majorization, and Hardy spaces’, Adv. Math. 26 (2) (1977), 182–205] imposed an additional condition on the exit time $\tau _{W}$ , namely the finiteness of $\mathbf {E}(\log (\tau _{W}))$ . Such a condition is typically difficult to verify, since the law of the exit time is often delicate. In this paper, we revisit Burkholder’s condition and propose an alternative viewpoint. Our approach is purely analytic, weaker and formulated in terms of proper analytic maps rather than exit times themselves. This provides a more flexible framework for further applications.
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