, Volume 135, Issue 4, pp 603-644
Date: 10 Nov 2005

Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space

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Abstract

In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic partial differential equation u(t,x)=1+ 0 t κΔ x u (s,x) ds+ 0 t W (ds,x) u (s,x), when the spatial parameter x is continuous, specifically xR, and W is a Gaussian field on R +×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the Lyapunov exponent defined as lim t →∞ t −1 log u(t,x). Furthermore, we find upper and lower bounds for lim sup t →∞ t −1 log u(t,x) and lim inf t →∞ t −1 log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal.

This author's research partially supported by NSF grant no. : 0204999