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

Ionuţ Florescu, Frederi Viens

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

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 + ∫0t κ Δxu (s, x) ds + ∫0t W (ds, x) u (s, x). when the spatial parameter x is continuous, specifically x ∈ R, 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 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.

Original languageEnglish
Pages (from-to)603-644
Number of pages42
JournalProbability Theory and Related Fields
Volume135
Issue number4
DOIs
StatePublished - Aug 2006

Keywords

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

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