Lifetime asymptotics of iterated brownian motion in ℝ n
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 147-160

Let τ D (Z) be the first exit time of iterated brownian motion from a domain D⊂ℝ n started at z∈D and let P z [τ D (Z)>t] be its distribution. In this paper we establish the exact asymptotics of P z [τ D (Z)>t] over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob. 14 (2004) 1529-1558] and Nane (2006) [Nane, Stochastic Processes Appl. 116 (2006) 905-916], for z∈D lim t→∞ t -1/2 exp3 2π 2/3 λ D 2/3 t 1/3 P z [τ D (Z)>t]=C(z), where C(z)=(λ D 2 7/2 )/3πψ(z)∫ D ψ(y)dy 2 . Here λ D is the first eigenvalue of the Dirichlet laplacian 1 2Δ in D, and ψ is the eigenfunction corresponding to λ D . We also study lifetime asymptotics of brownian-time brownian motion, Z t 1 =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.

DOI : 10.1051/ps:2007012
Classification : 60J65, 60K99
Keywords: iterated brownian motion, brownian-time brownian motion, exit time, bounded domain, twisted domain, unbounded convex domain 
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     title = {Lifetime asymptotics of iterated brownian motion in $\mathbb {R}^{n}$},
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     url = {https://www.numdam.org/articles/10.1051/ps:2007012/}
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Nane, Erkan. Lifetime asymptotics of iterated brownian motion in $\mathbb {R}^{n}$. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 147-160. doi: 10.1051/ps:2007012

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