Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions. (English) Zbl 1276.60023

Genadot, Alexandre; Thieullen, Michèle

Adv. Appl. Probab. 44, No. 3, 749-773 (2012).

The paper discusses a model reduction technique for certain piecewise deterministic processes (PDPs) taking values in Hilbert spaces. This type of càdlàg Markov process describes behaviour of a process with temporal evolution usually given by a (parabolic) partial differential equation interlaced with random jumps in its coefficients where the piecewise deterministic and the stochastic jump evolution are fully coupled. The designation ‘fully coupled’ means that on one hand the temporal evolution depends on the random jumps of the coefficient and on the other hand the random dynamics of the jumps depend on the deterministic temporal evolution. In the application of main interest is a stochastic spatio-temporal neuron model introduced in [T. D. Austin, Ann. Appl. Probab. 18, No. 4, 1279–1325 (2008; Zbl 1157.60066)], the model is a reaction-diffusion equation with randomly changing coefficients in the reaction term.
Fixing the inter-jump evolution to a constant value, the jumping part alone usually forms a continuous-time Markov chain. The authors then discuss the setup when the possible transitions in the Markov chain decompose in fast and slow jumps effectively becoming a slow-fast system. A model reduction is then possible for taking the fast subsystem to its limit which is given by the stationary solution to the fast sub-system Markov chain, hence the term ‘averaging’. In the study this limit is established in a mathematical rigorous way in the sense of weak convergence of the laws of the processes. The general theory developed for infinite dimensional PDPs is then applied to the axon model of [loc. cit.]. A numerical example illustrates that the averaged equation preserves the essential qualitative behaviour of the more detailed model.

Reviewer: Martin Riedler (Wien)

MSC:

60F05	Central limit and other weak theorems
60J75	Jump processes (MSC2010)
92C20	Neural biology
92C45	Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
47B80	Random linear operators

Keywords:

piecewise-deterministic Markov process; stochastic hybrid system; averaging principle; slow-fast system; neuron model; Hodgkin-Huxley model

Citations:

Zbl 1157.60066

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