Probability Theory and Related Fields

, Volume 135, Issue 4, pp 603-644

First online:

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

  • Ionuţ FlorescuAffiliated withDept. Statistics & Dept. Mathematics, Purdue University Email author 
  • , Frederi ViensAffiliated withDept. Statistics & Dept. Mathematics, Purdue University

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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.

Mathematics Subject Classification (2000)

primary 60H15 secondary 60G15 60H07

Keywords or phrases

Stochastic partial differential equations Anderson model Lyapunov exponent Gaussian regularity Malliavin calculus Feynman-Kac