陷阱模型的猝灭标度极限
Quenched scaling limits of trap models
摘要：
在$d$维环面$\bb T^d$上固定严格正测度$W$.对于整数$N\ge 1$，用$W^N_x$、$x=(x_1, ..., x_d)$、$0\le x_i <N$表示，$W$ -立方$[x/N, (x+\mb 1)/N)$的测度，其中$\mb 1$是所有分量都等于1的向量。在1维，我们证明了流体动力学行为的叠加的独立随机行走，其中一个粒子从$x/N$跳到它的邻居之一，在率$(N W^N_x)^{-1}$，描述的扩散标度的线性微分方程$\partial_t ρ= (d/dW)(d/dx) ρ$。在维度$d>1$，如果$W$是一个有限的离散措施，$W=\sum_{i\ge 1} w_i δ_{x_i}$，我们证明了随机游动跳跃从$x/N$均匀的速度$(W^N_x)^{-1}$其邻居之一，具有亚稳态行为，定义在\cite{bl1}，描述的$K$ -过程中引入\cite{fm1}。
英文摘要：
Fix a strictly positive measure $W$ on the $d$-dimensional torus $\bb T^d$. For an integer $N\ge 1$, denote by $W^N_x$, $x=(x_1, ..., x_d)$, $0\le x_i <N$, the $W$-measure of the cube $[x/N, (x+\mb 1)/N)$, where $\mb 1$ is the vector with all components equal to 1. In dimension 1, we prove that the hydrodynamic behavior of a superposition of independent random walks, in which a particle jumps from $x/N$ to one of its neighbors at rate $(N W^N_x)^{-1}$, is described in the diffusive scaling by the linear differential equation $\partial_t ρ= (d/dW)(d/dx) ρ$. In dimension $d>1$, if $W$ is a finite discrete measure, $W=\sum_{i\ge 1} w_i δ_{x_i}$, we prove that the random walk which jumps from $x/N$ uniformly to one of its neighbors at rate $(W^N_x)^{-1}$ has a metastable behavior, as defined in \cite{bl1}, described by the $K$-process introduced in \cite{fm1}.
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