On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators
On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators
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DOI：
10.1016/j.jde.2023.03.011
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发表时间：
2022-06
影响因子：
2.4
通讯作者：
Evan Randles;L. Saloff‐Coste
中科院分区：
文献类型：
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作者：
Evan Randles;L. Saloff‐Coste
我们考虑某些常系数微分算子的R d具有正定符号。每一个这样的符号为P的算子Λ定义了一个算子e− t Λ，t> 0的半群，允许一个连续的卷积核H P t，对于这个卷积核H P t（0）的大时间行为不能通过基本的标度参数推导出来。最简单的例子有符号P（n）=（η+ n 2）2+ η 4，n =（η，n）∈ R 2。我们设计了一种方法，使我们能够确定的大时间行为的HP t（0）的几类这种类型的例子，我们表明，这些大时间的渐近性是由某些高阶微分算子的扰动Λ保持。对于刚刚给出的P，当t趋于无穷大时，H P t（0）<$c P t− 5/8。我们展示了这些结果如何与理解Z d上某些有限支撑复函数的迭代卷积幂相关。我们还讨论了如何这些技术提供精确的小时间渐近的HPt（0）在某些情况下，算子Λ不是亚椭圆。最简单的例子Λ的符号为P（π）= η 2+（η− π 2）4，我们证明了在这种情况下，当t趋于0时，H P t（0）π c P t− 1/2。我们的工作代表了第一个基本的一步，很好地理解与这些微分算子的半群。获得这些半群的卷积核的有意义的非对角上界仍然是一个有趣的挑战。
We consider certain constant-coefficient differential operators on R d that have positive-definite symbols. Each such operator Λ with symbol P defines a semigroup of operators e− t Λ, t> 0, admitting a continuous convolution kernel H P t 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)∈ R 2. We devise a method that allows us to determine the large-time behavior of H P t (0) for several classes of examples of this type and we show that these large-time asymptotics are preserved by perturbations of Λ by certain higher-order differential operators. For the P just given, it turns out that H P t (0)∼ c P t− 5/8 when t tends to infinity. We show how such results are relevant to understand the iterated convolution powers of certain finitely-supported complex functions on Z d. We also discuss how these techniques provide precise small-time asymptotics for H P t (0) in some cases when the operator Λ is not hypoelliptic. The simplest such example Λ has symbol P (ξ)= η 2+(η− ξ 2) 4 and we show that H P t (0)∼ c P t− 1/2 as t tends to 0 in this case. Our work represents a first basic step towards a good understanding of the semigroups associated with these differential operators. Obtaining meaningful off-diagonal upper bounds for the convolution kernels of these semigroups remains an interesting challenge.