Iterated Brownian Motion in Parabola-Shaped Domains

Abstract

Iterated Brownian motion $Z_t$ serves as a physical model for diffusions in a crack. If ${\tau }_D(Z)$ is the first exit time of this processes from a domain $D \subset \RR{R}^{n}$, started at $z\in D$, then $P_z[{\tau }_D(Z)>t]$ is the distribution of the lifetime of the process in $D$. In this paper we determine the large time asymptotics of $P_z[{\tau }_{P_{\alpha }}(Z)>t]$ which gives exponential integrability of ${\tau }_{P_{\alpha }}(Z)$ for parabola-shaped domains of the form $ P_α=\{(x,Y)\in \RR{R} \times \RR{R}^{n-1}: x>0, |Y|<Ax^α \}$, for $0<\alpha <1$, $A>0.$ We also obtain similar results for twisted domains in $\RR{R}^{2}$ as defined in \cite{DSmits}. In particular, for a planar iterated Brownian motion in a parabola $\mathcal{P}=\{(x,y):x>0,|y|<\sqrt{x}\}$ we find that for $z\in \mathcal{P}$ $$\underset{t\to \infty }{\lim }{t}^{-1/7}\log P_z[{\tau }_{\mathcal{P}}(Z)>t]=-\frac{7{\pi }^2}{2^{25/7}}.$$

23 pages

References in corpus (1)

• Iterated Brownian motion in an open set

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