3 research outputs found

    Lifetime asymptotics of iterated Brownian motion in R^{n}

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    Let τD(Z)\tau_{D}(Z) be the first exit time of iterated Brownian motion from a domain D \subset \RR{R}^{n} started at zDz\in D and let Pz[τD(Z)>t]P_{z}[\tau_{D}(Z) >t] be its distribution. In this paper we establish the exact asymptotics of Pz[τD(Z)>t]P_{z}[\tau_{D}(Z) >t] over bounded domains as an improvement of the results in \cite{deblassie, nane2}, for zDz\in D \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 C(z)=(λD27/2)/3π(ψ(z)Dψ(y)dy)2C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}(\psi(z)\int_{D}\psi(y)dy) ^{2}. Here λD\lambda_{D} is the first eigenvalue of the Dirichlet Laplacian 1/2Δ{1/2}\Delta in DD, and ψ\psi is the eigenfunction corresponding to λD\lambda_{D} . We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM), Zt1=z+X(Y(t))Z^{1}_{t}=z+X(|Y(t)|), where XtX_{t} and YtY_{t} are independent one-dimensional Brownian motions

    Iterated Brownian motion in bounded domains in R^n

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    Let τD(Z)\tau_{D}(Z) is the first exit time of iterated Brownian motion from a domain D \subset \RR{R}^{n} started at zDz\in D and let Pz[τD(Z)>t]P_{z}[\tau_{D}(Z) >t] be its distribution. In this paper we establish the exact asymptotics of Pz[τD(Z)>t]P_{z}[\tau_{D}(Z) >t] over bounded domains as an extension of the result in DeBlassie \cite{deblassie}, for zDz\in D Pz[τD(Z)>t]t1/2exp(3/2π2/3λD2/3t1/3),ast. P_{z}[\tau_{D}(Z)>t]\approx t^{1/2} \exp(-{3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}), as t\to\infty . We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), Zt1=z+X(Y(t))Z^{1}_{t}=z+X(Y(t)), where XtX_{t} and YtY_{t} are independent one-dimensional Brownian motions.Comment: 17 page
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