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Lifetime asymptotics of iterated Brownian motion in R^{n}
Let be the first exit time of iterated Brownian motion from a
domain D \subset \RR{R}^{n} started at and let be its distribution. In this paper we establish the exact asymptotics of
over bounded domains as an improvement of the results
in \cite{deblassie, nane2}, for \begin{eqnarray}
\lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3})
P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where
. Here
is the first eigenvalue of the Dirichlet Laplacian
in , and is the eigenfunction corresponding to .
We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM),
, where and are independent
one-dimensional Brownian motions
Hardy-Stein identities and square functions for semigroups
We prove a Hardy-Stein type identity for the semigroups of symmetric,
pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it
gives the two-way boundedness, for , of the corresponding
Littlewood-Paley square function. The square function yields a direct proof of
the boundedness of Fourier multipliers obtained by transforms of
martingales of L\'evy processes.Comment: 17 page
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