We study the differential operator \(A=\frac{d^4}{dx^4}\) acting on a connected network \(\mathcal {G}\) along with \(\mathcal L^2\), the square of the discrete Laplacian acting on a connected discrete graph \(\mathsf {G}\). For both operators, we discuss well-posedness of the associated linear parabolic problems$$\begin{aligned} \frac{\partial u}{\partial t}=-Au,\qquad \frac{df}{dt}=-{\mathcal {L}}^2 f, \end{aligned}$$on \(L^p(\mathcal {G})\) or \(\ell ^p(\mathsf {V})\), respectively, for \(1\le p\le \infty \). In view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order 2N for \(N>1\), our most surprising finding is that after some transient time, the parabolic equations driven by \(-A\) may display Markovian features, depending on the imposed transmission conditions in the vertices. Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by \(-{\mathcal {L}}^2\) is also presented.

中文翻译：

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图和网络上的双Laplacians

我们研究作用于连接网络\（\ mathcal {G} \）以及\（\ mathcal L ^ 2 \）的微分算子\（A = \ frac {d ^ 4} {dx ^ 4} \），作用于相连离散图\（\ mathsf {G} \）上的离散拉普拉斯算子的平方。对于这两个算子，我们讨论相关的线性抛物线问题的适定性$$ \ begin {aligned} \ frac {\ partial u} {\ partial t} =-Au，\ qquad \ frac {df} {dt} =- {\ mathcal {L}} ^ 2 f，\ end {aligned} $$ on \（L ^ p（\ mathcal {G}）\）或\（\ ell ^ p（\ mathsf {V}）\），分别用于\（1 \ le p \ le \ infty \）。鉴于众所周知所有2阶椭圆微分算子都没有抛物线最大原理Ñ为\（N> 1 \） ，我们最惊人的发现是，一些过渡时间之后，通过驱动的抛物线方程\（ - A \）可以显示马尔可夫特征，根据在顶点施加的传输条件。在一般域甚至有界区间的情况下，类似结果似乎是未知的。我们的分析基于对双谐波函数的详细研究，并辅以简单的组合论证。我们详细阐述了离散双拉普拉斯算子的类似问题。根据\（-{\ mathcal {L}} ^ 2 \）生成的半群的马尔可夫性质，给出了完整图的表征。