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Abstract

Let $\tau$ be the first hitting time of the point $1$ by the geometric Brownian motion $X(t)= x \exp(B(t)-2\mu t)$ with drift $\mu \geq 0$ starting from $x>1$. Here $B(t)$ is the Brownian motion starting from $0$ with $E B^2(t) = 2t$. We provide an integral formula for the density function of the stopped exponential functional $A(\tau)=\int_0^\tau X^2(t)\, dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in \cite{BGS}, the present paper covers the case of arbitrary drifts $\mu \geq 0$ and provides a significant unification and extension of the results of the above-mentioned paper. As a corollary we provide an integral formula and give the asymptotic behaviour at infinity of the Poisson kernel for half-spaces for Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.