Quenched to annealed transition in the parabolic Anderson problem

• 成果类型: Article
• 署名作者: Cranston, M.; Molchanov, S.
• 署名单位: University of California System; University of California Irvine; University of North Carolina; University of North Carolina Charlotte
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-006-0020-7
• 发表日期: 2007
• 页码: 177-193

摘要

We study limit behavior for sums of the form 1/vertical bar Lambda(L)vertical bar Sigma(x is an element of Lambda L) u(t,x), where the field {u(t,x) : x is an element of Z(d)} is composed of solutions of the parabolic Anderson equation u (t,X) = 1 + kappa integral(t)(0) Delta u(s,x) ds + integral(t)(o) u(s,x) partial derivative B-x (S). The index set is a box in Z(d), namely Lambda(L) = {x is an element of Z(d) : vertical bar x vertical bar <= L} and L = L(t) is a nondecreasing function L : [0, infinity) -> R+. We identify two critical parameters eta(1) < eta(2) such that for gamma > eta(1) and L(t) = e(gamma t), the sums 1/vertical bar Lambda(L)vertical bar Sigma(x is an element of Lambda L) u(t,x) satisfy a law of I arge numbers, or put another way, they exhibit annealed behavior. For gamma > eta(2) and L(t) = e(gamma t), one has Sigma(x is an element of Lambda L) u(t,x) when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when lim(1 ->infinity) 1/t ln L(t) = 0, quenched asymptotics occur. That means lim(t ->infinity) ln (1/vertical bar Lambda(L)vertical bar Sigma(x is an element of Lambda L) u(t,x) = gamma (kappa), where gamma (kappa) is the almost sure Lyapunov exponent, i.e. lim(t ->infinity) 1/t ln u(t,x) = gamma(kappa). We also examine the behavior of 1/vertical bar Lambda(L)vertical bar Sigma(x is an element of Lambda L) u(t, x) for L = e(gamma t) with gamma in the transition range (0, eta(1)).