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      Lyapunov Exponents of Brownian Motion: Decay Rates for Scaled Poissonian Potentials and Bounds

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          Abstract

          We investigate Lyapunov exponents of Brownian motion in a nonnegative Poissonian potential \(V\). The Lyapunov exponent depends on the potential \(V\) and our interest lies in the decay rate of the Lyapunov exponent if the potential \(V\) tends to zero. In our model the random potential \(V\) is generated by locating at each point of a Poisson point process with intensity \(\nu\) a bounded compactly supported nonnegative function \(W\). We show that for sequences of potentials \(V_n\) for which \(\nu_n \|W_n\|_1 \sim D/n\) for some constant \(D > 0\) (\(n \to \infty\)), the decay rates to zero of the quenched and annealed Lyapunov exponents coincide and equal \(c n^{-1/2}\) where the constant \(c\) is computed explicitly. Further we are able to estimate the quenched Lyapunov exponent norm from above by the corresponding norm for the averaged potential.

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          Most cited references9

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          Foundations of Modern Probability

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            Shape theorem, lyapounov exponents, and large deviations for brownian motion in a poissonian potential

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              Large deviations and phase transition for random walks in random nonnegative potentials

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                Author and article information

                Journal
                18 January 2011
                2011-10-19
                Article
                1101.3404
                4157df45-36a0-4e0b-9ff6-9157bf1b20b0

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                60K37, 60J65 (Primary), 82B44 (Secondary)
                Now 14 pages, 2 figures. Some references added, abstract changed, 2 new paragraphs in the introduction
                math.PR

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