arXiv 1407.32402014-07math.PR

Liouville热核的连续性和估计及其在谱维数上的应用

Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions

Sebastian Andres, Naotaka Kajino

备注：: 36 pages

摘要：

刘维尔布朗运动（Liouville Brownian motion，LBM）是由Garban、Rhodes和Vargas最近提出的，也是Berestycki提出的一种较弱形式，是由刘维尔测度$M_γ$诱导的平面随机几何中的扩散过程，对于（大质量）高斯自由场$X$，正式写作$M_γ(dz)=e^{γX(z)-{γ^2} \mathbb{E}[X(z)^2]/2}\, dz$，$γ\in(0,2)$。它是一个$M_γ$ -对称扩散，定义为二维布朗运动的时间变化的正连续加性泛函与Revuz测度$M_γ$。在本文中，我们提供了一个详细的分析热核$p_t(x,y)$的LBM。具体来说，我们证明了它的联合连续性，局部均匀的次高斯上限的形式$p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}((|x-y|^β\wedge 1)/t)^{\frac{1}{β-1}}\bigr)$的$t\in(0,\frac{1}{2}]$为每个$β>\frac{1}{2}(γ+2)^2$，和对角下界的形式$p_{t}(x,x)\geq C_{3}t^{-1}\bigl(\log(t^{-1})\bigr)^{-η}$的$t\in(0,t_η(x)]$，与$t_η(x)\in(0,\frac{1}{2}]$严重依赖于$x$，为每个$η>18$的$M_γ$ -几乎每一个$x$。作为应用，我们推导出点态谱维数等于$2$ $M_γ$ -a.e.并且全局谱维数也是$2$。

英文摘要：

The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $M_γ$, formally written as $M_γ(dz)=e^{γX(z)-{γ^2} \mathbb{E}[X(z)^2]/2}\, dz$, $γ\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_γ$-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_γ$. In this paper we provide a detailed analysis of the heat kernel $p_t(x,y)$ of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}((|x-y|^β\wedge 1)/t)^{\frac{1}{β-1}}\bigr)$ for $t\in(0,\frac{1}{2}]$ for each $β>\frac{1}{2}(γ+2)^2$, and an on-diagonal lower bound of the form $p_{t}(x,x)\geq C_{3}t^{-1}\bigl(\log(t^{-1})\bigr)^{-η}$ for $t\in(0,t_η(x)]$, with $t_η(x)\in(0,\frac{1}{2}]$ heavily dependent on $x$, for each $η>18$ for $M_γ$-almost every $x$. As applications, we deduce that the pointwise spectral dimension equals $2$ $M_γ$-a.e.\ and that the global spectral dimension is also $2$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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