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Lifetime asymptotics of iterated Brownian motion in \(\mathbb R^n\). (English) Zbl 1181.60127

Summary: Let \(\tau _{D}(Z) \) be the first exit time of iterated Brownian motion from a domain \(D \subset \mathbb{R}^{n}\) started at \(z\in D\) and let \(P_{z}[\tau _{D}(Z) >t]\) be its distribution. In this paper we establish the exact asymptotics of \(P_{z}[\tau _{D}(Z) >t]\) over bounded domains as an improvement of the results in R. D. DeBlassie [Ann. Appl. Probab. 14, No. 3, 1529–1558 (2004; Zbl 1051.60082 )] and E. Nane [Stochastic Processes Appl. 116, No. 6, 905–916 (2006; Zbl 1106.60309)], for \(z\in D\) \[ \lim _{t\rightarrow \infty } t^{-1/2}\exp \left(\frac{3}{2}\pi ^{2/3}\lambda _{D}^{2/3}t^{1/3}\right) P_{z}[\tau _{D}(Z)>t]= C(z), \] where \(C(z)=(\lambda _{D}2^{7/2})/\sqrt{3 \pi }\left( \psi (z)\int _{D}\psi (y)\text{d}y\right) ^{2}\). Here \(\lambda _{D}\) is the first eigenvalue of the Dirichlet laplacian \(\frac{1}{2}\Delta \) in \(D\), and \(\psi \) is the eigenfunction corresponding to \(\lambda _{D}\). We also study lifetime asymptotics of Brownian-time Brownian motion, \(Z^{1}_{t} = z+X(|Y(t)|)\), where \(X_{t}\) and \(Y_{t}\) are independent one-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.

MSC:

60J65 Brownian motion
60K99 Special processes
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