Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks

TL;DR

This paper develops analytical criteria for viability, invariance, and reachability in controlled piecewise deterministic Markov processes, with applications to gene network models using viscosity solutions and duality methods.

Contribution

It introduces new viscosity-based criteria for viability and invariance, and applies these to biological gene network models, advancing the understanding of controlled PDMPs.

Findings

Established viability and invariance criteria using viscosity solutions.

Analyzed reachability of open sets in controlled PDMPs.

Applied theoretical results to biological gene network models.

Abstract

We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary open sets. The method is based on viscosity techniques and duality for some associated linearized problem. The theoretical results are applied to general On/Off systems, Cook's model for haploinssuficiency, and a stochastic model for bacteriophage lambda.