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

Article

DOI: 10.1007/s00440-005-0471-2

Cite this article as:
Florescu, I. & Viens, F. Probab. Theory Relat. Fields (2006) 135: 603. doi:10.1007/s00440-005-0471-2

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+0tκΔxu(s,x)ds+0tW(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 limt→∞t−1 log u(t,x). Furthermore, we find upper and lower bounds for lim supt→∞t−1 log u(t,x) and lim inft→∞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.

Mathematics Subject Classification (2000)

primary 60H15secondary 60G1560H07

Keywords or phrases

Stochastic partial differential equationsAnderson modelLyapunov exponentGaussian regularityMalliavin calculusFeynman-Kac

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Dept. Statistics & Dept. MathematicsPurdue UniversityWest LafayetteUSA