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Let ${\tau }_D\left(Z\right)$ be the first exit time of iterated Brownian motion from a domain $D\subset {ℝ}^n$ started at $z\in D$ and let $P_z[{\tau }_D\left(Z\right)>t]$ be its distribution. In this paper we establish the exact asymptotics of $P_z[{\tau }_D\left(Z\right)>t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, (2004) 1529–1558] and Nane (2006) [Nane, (2006) 905–916], for $z\in D$
 ${\displaystyle \underset{t\to \infty }{lim}{t}^{-1/2}exp\left(\frac{3}{2}{\pi }^{2/3}{\lambda }_D^{2/3}{t}^{1/3}\right)P_z[{\tau }_D\left(Z\right)>t]=C\left(z\right),}$
where $C\left(z\right)=\left({\lambda }_D{2}^{7/2}\right)/\sqrt{3\pi }{\left(\psi \left(z\right){\int }_D\psi \left(y\right)\mathrm{d}y\right)}^2$. Here is the first eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ in , and is the eigenfunction corresponding to . We also...

For 0 <  ≤ 2 and 0 <  < 1, an -time fractional Brownian motion is an iterated process  =  {() = (()) ≥ 0}  obtained by taking a fractional Brownian motion  {() ∈ ℝ} with Hurst index 0 <  < 1 and replacing the time parameter with a strictly -stable Lévy process {() ≥ 0} in ℝ independent of {() ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when is a stable subordinator, can arise as scaling limit of randomly...

For 0 <  ≤ 2 and 0 <  < 1, an -time fractional Brownian motion is an iterated process  =  {() = (()) ≥ 0}  obtained by taking a fractional Brownian motion  {() ∈ ℝ} with Hurst index 0 <  < 1 and replacing the time parameter with a strictly -stable Lévy process {() ≥ 0} in ℝ independent of {() ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when ...