AI 中文总结
本研究证明了$\u2124^2$上有限方差非周期对称随机游走的最优Hardy不等式，验证了Hardy权的零临界性与最优性，推导了权的渐近式，给出了常数上界，还提出了一般图的零临界性新判据并可推广至高维格点情形。
AI 中文摘要
我们证明了$\u2124^2$上所有具有有限方差的非周期对称随机游走的最优Hardy不等式。特别地，我们验证了零临界性，从而验证了底层Hardy权的最优性。在合适的矩条件下，我们利用Fukai和Uchiyama提出的位势核的精细渐近式，推导了该权的渐近行为。对于标准拉普拉斯算子，我们复现了渐近式中预期的一阶项，同时还证明了下一阶项为负。因此，我们的结果表明，Kapitanski和Laptev证明的Hardy不等式中的常数不能大于$1/4$，这是连续情形下的最优常数。零临界性的证明依赖于一个适用于超越局部有限情形的一般图的新判据。我们还复现了$d\geq3$时$\u2124^d$的情形，该情形也可以用我们的新方法处理。
英文摘要
We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.
Comments28 pages, comments are welcome
