Piecewise-deterministic Markov processes as limits of Markov jump processes. (English) Zbl 1254.60031

Franz, Uwe; Liebscher, Volkmar; Zeiser, Stefan

Adv. Appl. Probab. 44, No. 3, 729-748 (2012).

Summary: A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper, we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling, the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology, where entities often occur in different scales of numbers.

MSC:

60F15	Strong limit theorems
60J28	Applications of continuous-time Markov processes on discrete state spaces
92C40	Biochemistry, molecular biology

Keywords:

piecewise-deterministic Markov process; stochastic model of gene regulation; limit theorem; Skorokhod space

References:

[1]	Ball, K., Kurtz, T. G., Popovic, L. and Rempala, G. (2006). Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Prob. 16 , 1925-1961. · Zbl 1118.92031
[2]	Becskei, A. and Serrano, L. (2000). Engineering stability in gene networks by autoregulation. Nature 405 , 590-593.
[3]	Blake, W. J., Kærn, M., Cantor, C. R. and Collins, J. J. (2003). Noise in eukaryotic gene expression. Nature 422 , 633-637.
[4]	Bundschuh, R., Hayot, F. and Jayaprakash, C. (2003). Fluctuations and slow variables in genetic networks. Biophys. J. 84 , 1606-1615.
[5]	Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46 , 353-388. · Zbl 0565.60070
[6]	Doob, J. L. (1953). Stochastic Processes . John Wiley, New York. · Zbl 0053.26802
[7]	Gillespie, D. T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22 , 403-434.
[8]	Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 , 2340-2361.
[9]	Hume, D. A. (2000). Probability in transcriptional regulation and its implications for leukocyte differentiation and inducible gene expression. Blood 96 , 2323-2328.
[10]	Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7 , 49-58. · Zbl 0191.47301
[11]	Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8 , 344-356. · Zbl 0219.60060
[12]	Kurtz, T. G. (1980). Relationships between stochastic and deterministic population models. In Biological Growth and Spread (Proc. Conf. Heidelberg, 1979; Lecture Notes Biomath. 38 ), Springer, Berlin, pp. 449-467. · Zbl 0471.92014
[13]	Maamar, H. and Dubnau, D. (2005). Bistability in the Bacillus subtilis K-state (competence) system requires a positive feedback loop. Mol. Microbiol. 56 , 615-624.
[14]	Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory Prob. Appl. 1 , 261-290. · Zbl 0074.33802
[15]	Zeiser, S., Franz, U. and Liebscher, V. (2009). Autocatalytic genetic networks modeled by piecewise-deterministic Markov processes. J. Math. Biol. 60 , 207-246. · Zbl 1311.92076
[16]	Zeiser, S., Franz, U., Müller, J. and Liebscher, V. (2009). Hybrid modeling of noise reduction by a negatively autoregulated system. Bull. Math. Biol. 71 , 1006-1024. · Zbl 1163.92028
[17]	Zeiser, S., Franz, U., Wittich, O. and Liebscher, V. (2008). Simulation of genetic networks modelled by piecewise deterministic Markov processes. IET Systems Biol. 2 , 113-135.