Long-time tails in the parabolic Anderson model with bounded potential
Long-time tails in the parabolic Anderson model with bounded potential
复制标题
具有有限潜力的抛物线安德森模型中的长期尾部
DOI：
10.1214/aop/1008956688
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发表时间：
2000
影响因子：
2.3
通讯作者：
Wolfgang Koenig
中科院分区：
文献类型：
--
作者：
Marek Biskup;Wolfgang Koenig
本文考虑（0，∞）× Zd上具有随机独立同分布的抛物型安德森问题u = κΔu + λ u.势函数ε =（ε（z））z ∈ Zd和初始条件u（0，.）2001年。我们的主要假设是esssup（0）= 0。根据分布的厚度，Prob（0）∈.在本质上确界附近，我们用变分问题的方法证明了u（t，0）的矩的渐近性和u（t，0）在t → ∞时的几乎必然渐近性.作为副产品，我们在随机薛定谔算子- κΔ - ξ的谱底部建立了Lifshitz尾。在我们的分布类中，Lifshitz指数的范围从d/2到∞;幂律通常伴随着低阶修正。
We consider the parabolic Anderson problem ∂ t u = κΔu + ξu on (0, ∞) × Z d with random i.i.d. potential ξ = (ξ(z)) z ∈ Zd and the initial condition u(0,.) ≡ 1. Our main assumption is that esssup ξ(0) = 0. Depending on the thickness of the distribution Prob(ξ(0) ∈.) close to its essential supremum, we identify both the asymptotics of the moments of u(t, 0) and the almost-sure asymptotics of u(t, 0) as t → ∞ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrodinger operator - κΔ - ξ at the bottom of its spectrum. In our class of distributions, the Lifshitz exponent ranges from d/2 to ∞; the power law is typically accompanied by lower-order corrections.