论文标题
一类不均匀的部分差分算子的热核上划分的渐近渐近线
On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators
论文作者
论文摘要
我们将某些具有正定符号的$ \ mathbb {r}^d $上的恒定差分运算符。每个此类操作员$λ$带有符号$ p $定义一个半群$ e^{ - tλ} $，$ t> 0 $，承认卷积内核$ h^t_p $为$ h_p^t（0）$的大型行为无法通过基本缩放量表参数来划分。最简单的示例具有符号$ p（ξ）=（η+ζ^2）^2+η^4 $，$ξ=（η，ζ）\ in \ mathbb {r}^2 $。我们设计了一种方法来建立这种类型的几类示例的$ h^t_p（0）$的大渐近学，我们表明这些渐近学是由某些高阶差异操作员通过扰动来保留的。对于仅给出的$ p $，事实证明$ h^t_p（0）\ sim c_pt^{ - 5/8} $ as $ t \ to \ to \ infty $。我们展示了这些结果与了解$ \ Mathbb z^d $上某些复杂函数的卷积能力是如何相关的。我们的工作是朝着对与这些运营商相关的半群的充分理解的第一步。为$ H_P^t $获得有意义的非对角线上限仍然是一个有趣的挑战。
We consider certain constant-coefficient differential operators on $\mathbb{R}^d$ with positive-definite symbols. Each such operator $Λ$ with symbol $P$ defines a semigroup $e^{-tΛ}$ , $t>0$ , admitting a convolution kernel $H^t_P$ for which the large-time behavior of $H_P^t(0)$ cannot be deduced by basic scaling arguments. The simplest example has symbol $P(ξ)=(η+ζ^2)^2+η^4$ , $ξ=(η,ζ)\in \mathbb{R}^2$ . We devise a method to establish large-time asymptotics of $H^t_P(0)$ for several classes of examples of this type and we show that these asymptotics are preserved by perturbations by certain higher-order differential operators. For the $P$ just given, it turns out that $H^t_P(0)\sim c_Pt^{-5/8}$ as $t\to\infty$ . We show how such results are relevant to understand the convolution powers of certain complex functions on $\mathbb Z^d$ . Our work represents a first basic step towards a good understanding of the semigroups associated with these operators. Obtaining meaningful off-diagonal upper bounds for $H_P^t$ remains an interesting challenge.
