Povel, Tobias The one dimensional annealed \(\delta\)-Lyapounov exponent. (English) Zbl 0903.60093 Ann. Inst. Henri Poincaré, Probab. Stat. 34, No. 1, 61-72 (1998). 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 \(\beta_\delta(c)\). Cited in 2 Documents MSC: 60K40 Other physical applications of random processes 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) Keywords:Brownian motion; Poisson points PDFBibTeX XMLCite \textit{T. Povel}, Ann. Inst. Henri Poincaré, Probab. Stat. 34, No. 1, 61--72 (1998; Zbl 0903.60093) Full Text: DOI Numdam EuDML