Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach

TL;DR

This paper establishes necessary regularity conditions for coefficients to ensure invariance and border avoidance in stochastic systems, with applications to gene networks and bacteriophage models, highlighting the importance of explicit environmental modeling.

Contribution

It provides a necessary local Lipschitz-like condition for invariance, extends border avoidance results to stochastic processes, and demonstrates the need for explicit environmental cues in biological models.

Findings

Necessary Lipschitz-like condition for invariance.

Border avoidance results for controlled diffusions and PDsMP.

Explicit environmental modeling is crucial for accurate biological predictions.

Abstract

Motivated by the result of invariance of regular-boundary open sets in \cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optimality are provided via examples. Second, we provide a border avoidance (near-viability) counterpart of \cite{CannarsaDaPratoFrankowska2009} for controlled Brownian diffusions and piecewise deterministic switched Markov processes (PDsMP). We equally discuss to which extent Lipschitz-continuity of the driving coefficients is needed. Finally, by applying the theoretical result on PDsMP to Hasty's model of bacteriophage (\cite{hasty\_pradines\_dolnik\_collins\_00}, \cite{crudu\_debussche\_radulescu\_09}), we show the necessity of explicit modeling for the environmental cue triggering lysis.