We extend the so-called lower-bound technique for equicontinuous families of Markov operators by introducing the new concept of uniform equicontinuity on balls. Combined with a semi-concentrating condition, it yields a new abstract mathematical result on existence and uniqueness of invariant measures for Markov operators. It allows us to show the tightness of the set of invariant measures for some classes of Markov operators. This, in turn, gives a useful tool for proving a continuous dependence on given parameters for semi–concentrating Markov semigroups. In the second part we formulate an abstract modelling framework that defines a piecewise deterministic Markov process whose transition operator at the times of intervention yields a semi-concentrating Markov operator that is uniformly equicontinuous on balls. We show that this framework applies to a detailed stochastic model for an autoregulated gene in a bacterium that takes random transcription delay into account.
Mots clés : Regulatory network model, Markov operators, invariant measure
Sander Hille 1 ; Katarzyna Horbacz 2 ; Tomasz Szarek 3
@article{AMBP_2016__23_2_171_0,
author = {Sander Hille and Katarzyna Horbacz and Tomasz Szarek},
title = {Existence of a unique invariant measure for a class of equicontinuous {Markov} operators with application to a stochastic model for an autoregulated gene},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {171--217},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {23},
number = {2},
year = {2016},
doi = {10.5802/ambp.360},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.360/}
}
TY - JOUR AU - Sander Hille AU - Katarzyna Horbacz AU - Tomasz Szarek TI - Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene JO - Annales mathématiques Blaise Pascal PY - 2016 SP - 171 EP - 217 VL - 23 IS - 2 PB - Annales mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.360/ DO - 10.5802/ambp.360 LA - en ID - AMBP_2016__23_2_171_0 ER -
%0 Journal Article %A Sander Hille %A Katarzyna Horbacz %A Tomasz Szarek %T Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene %J Annales mathématiques Blaise Pascal %D 2016 %P 171-217 %V 23 %N 2 %I Annales mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.360/ %R 10.5802/ambp.360 %G en %F AMBP_2016__23_2_171_0
Sander Hille; Katarzyna Horbacz; Tomasz Szarek. Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene. Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 171-217. doi : 10.5802/ambp.360. https://ambp.centre-mersenne.org/articles/10.5802/ambp.360/
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