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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.

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