Self-adjoint and Markovian extensions of infinite quantum graphs.

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, , and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self-adjointness of the Gaffney Laplacian - the underlying metric graph does not have finite volume ends.

Analytic solutions for stochastic hybrid models of gene regulatory networks.

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative.

Stochastic hybrid models of gene regulatory networks - A PDE approach.

A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation approach based on a system of partial differential equations, where we assume a continuous-deterministic evolution for the protein counts.