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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)
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