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Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. (English) Zbl 0818.60071
Consider \(P_ X\), the probability measure on Wiener space such that the coordinate map \(B_ t(w) = w(t)\) is a Brownian motion starting at \(x > 0\). For \(\alpha > 0\) consider the Brownian motion with negative drift \(X_ t = B_ t - \alpha t\) and denote the hitting time of 0 by \(\tau^ X_ 0 = \inf \{t > 0 : X_ t = 0\}\). By \(P_ t^{(\alpha)}\) denote the sub-Markovian semigroup \(P_ t^{(\alpha)} \psi (x) = \mathbb{E}_ X (\psi (X_ t), \tau^ X_ 0 > t)\) and by \(p^{(\alpha)} (t,x,y)\) its transition density. For the sub-Markovian semigroup \((P^ \sim_ t)\) a measure \(\mu\) is quasi-invariant if there exists a real \(c\) such that \(\mu P^ \sim_ t = e^{ct} \mu\) for any \(t\). If in addition \(\mu\) is a probability measure, then it is called a quasi-stationary distribution. It is proved that the quasi-invariant measures associated to a Brownian motion with negative drift \(X\) form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three- dimensional Bessel process, and it is shown that it is the density of the limit distribution \(\lim \mathbb{P}_ X (X \in A \mid \tau^ X_ 0 > t)\) when \(t \to \infty\). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution \(\lim \mathbb{P}_ X (X \in A \mid \tau^ X_ 0 > \tau^ X_ M)\) when \(M \to \infty\) which is the law of a Bessel process with drift.

60J65 Brownian motion
60G50 Sums of independent random variables; random walks
60F99 Limit theorems in probability theory
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