We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

We prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces: if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete. We use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure: if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete, where the cubic volume growth is sharp.

No abstract

Endothelial permeability has been extensively investigated in the context of pathologies such as cancer and also in studies of drug delivery from the circulation. Hypoxia is a critical regulator of endothelial cell (EC) behavior and affects the barrier function of endothelial linings, yet its role has been little studied. This paper reveals the effect of hypoxia on the permeability of an EC monolayer by cellular experiments using a microfluidic device and a conventional cell culture dish. Human umbilical vein endothelial cells (HUVECs) were seeded into one microfluidic channel, creating an EC monolayer on each vertical surface of a collagen gel confined to a central chamber. Oxygen tension was regulated to produce normoxic (21% O) or hypoxic (3% O) conditions by the supply of gas mixtures of oxygen, carbon dioxide, and nitrogen at predefined ratios into channels fabricated into the device. Permeability of the EC monolayer quantified by analyzing diffusion of fluorescence-labelled dextrans into the collagen gel increases with barrier function loss by 6 hour hypoxic exposure, showing 11-fold and 4-fold increases for 70 kDa and 10 kDa dextrans, respectively, on average. Consistent with this, subsequent immunofluorescent staining and separate western blot analysis of HUVECs on a culture dish demonstrate loose cell-cell adhesion resulting from internalization of VE-cadherin under hypoxia. Thus, hypoxic stress increases endothelial permeability by altering cell-cell junction integrity.