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The one dimensional annealed $$\delta$$-Lyapounov exponent. (English) Zbl 0903.60093
Summary: We study a one-dimensional Brownian motion moving among Poisson points of constant intensity $$\nu>0$$. We introduce the “annealed $$\delta$$-Lyapunov exponent” $$\beta_\delta(c)$$. Here “annealed” refers to the fact that averages are both taken with respect to the path and environment measures. The exponent $$\beta_\delta(c)$$ measures how costly it is for the Brownian motion to reach a remote location while it receives a penalty “proportional” to $$c\in(0,\infty)$$ for spending too much time at Poisson points, and when the particle can pick its own time to perform the displacement. We derive a formula for $$\beta_\delta(c)$$, which shows that for all $$c\in(0,\infty)$$, $$\beta_\delta(c)<\nu$$. We conjecture that in general this is also true for the one-dimensional “annealed Lyapunov exponent”, introduced by Sznitman, which is an analogue object to $$\beta_\delta(c)$$.

##### MSC:
 60K40 Other physical applications of random processes 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
##### Keywords:
Brownian motion; Poisson points
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