TY - JOUR
T1 - Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space
AU - Florescu, Ionuţ
AU - Viens, Frederi
PY - 2006/8
Y1 - 2006/8
N2 - 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.
AB - 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.
KW - Anderson model
KW - Feynman-Kac
KW - Gaussian regularity
KW - Lyapunov exponent
KW - Malliavin calculus
KW - Stochastic partial differential equations
UR - http://www.scopus.com/inward/record.url?scp=33646753583&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=33646753583&partnerID=8YFLogxK
U2 - 10.1007/s00440-005-0471-2
DO - 10.1007/s00440-005-0471-2
M3 - Article
AN - SCOPUS:33646753583
SN - 0178-8051
VL - 135
SP - 603
EP - 644
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
IS - 4
ER -