发表机构
University of Science and Technology of China; University of Manchester（中国科学技术大学; 曼彻斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立了随机环境中一维扩散（含Brox扩散）淬火与退火热核的Varadhan小时间渐近公式，极限由内蕴坐标距离决定，并在紧集上一致成立。
AI 中文摘要
我们建立了随机环境中一维扩散的淬火和退火热核的Varadhan小时间渐近性，其生成元为$\mathcal L_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$。系数$a$和$\rho$在空间上连续，并满足局部指数矩条件。我们假设内蕴坐标映射$\Lambda_W$的分布具有紧支撑$\mathscr L$，该支撑由严格递增函数组成。设$q^W(t,x,y)$和$q(t,x,y)=\mathbb E[q^W(t,x,y)]$分别表示关于勒贝格测度的淬火和退火热核。我们证明，对于几乎每个环境$W$，$\lim_{t\downarrow0}t\log q^W(t,x,y)=-\frac12|\Lambda_W(y)-\Lambda_W(x)|^2$，且$\lim_{t\downarrow0}t\log q(t,x,y)=-\frac12\min_{\Lambda\in\mathscr L}|\Lambda(y)-\Lambda(x)|^2$。两个极限在$\mathbb R^2$的紧子集上一致成立。该框架包括Brox扩散，其形式描述为$dX_t=dB_t-\frac12\dot W(X_t)\\,dt$，其中$B$是标准布朗运动，$W$是独立的两边布朗运动。
英文摘要
We establish the Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator $\mathcal L_W f(x)=e^{-ρ(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $a$ and $ρ$ are continuous in space and satisfy a local exponential moment condition. We assume that the law of the intrinsic coordinate map $Λ_W$ has compact support $\mathscr L$ consisting of strictly increasing functions. Let $q^W(t,x,y)$ and $q(t,x,y)=\mathbb E[q^W(t,x,y)]$ denote the quenched and annealed heat kernels with respect to Lebesgue measure, respectively. We prove that, for almost every environment $W$, $\lim_{t\downarrow0}t\log q^W(t,x,y)=-\frac12|Λ_W(y)-Λ_W(x)|^2$, and that $\lim_{t\downarrow0}t\log q(t,x,y)=-\frac12\min_{Λ\in\mathscr L}|Λ(y)-Λ(x)|^2$. Both limits hold uniformly on compact subsets of $\mathbb R^2$. The framework includes Brox diffusion, formally described by $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion.
Comments13 pages
