论文标题
具有白噪声潜力的二维抛物线型安德森模型的长期渐近学
Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
论文作者
论文摘要
我们考虑抛物线型安德森模型（PAM）$ \ partial_t u = \frac12ΔU +ξu$ in $ \ mathbb r^2 $，带有高斯（太空）白色潜在$ξ$。我们证明，在时间$ t $，书面$ u（t）$的总质量的几乎纯净的大渐近行为是由$ \ log u（t）\ simχt\ simχt\ log t $ for $ t \ for $ t \ to \ infty $的，确定性常量$χ$χ$以各种公式确定。在其中一位作者的早期工作中，该常数用于描述主要eigenvalue $ \boldsymbolλ_1（q_t）的渐近行为$ \boldsymbolλ_1（q_t）\simχ\ log t $ t $boldsymbolλ_1（q_t）$ a anderson Operator的box $ q_t = $ q_t = $ q_t = = q_t = = q_t = = q_t = = q_t = = q_t = = q_t = = = q_t = = = = [ - \ frac {t} {2}，\ frac {t} {2}]^2 $。
We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 Δu + ξu$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $ξ$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $\log U(t)\sim χt \log t$ for $t \to \infty$, with the deterministic constant $χ$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour $\boldsymbol λ_1(Q_t)\simχ\log t$ of the principal eigenvalue $\boldsymbolλ_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t= [-\frac{t}{2},\frac{t}{2}]^2$.
