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**A simple stochastic kinetic transport model.**
*(English)*
Zbl 1264.60053

The authors propose to connect a stochastic single-particle model to the well-known deterministic model which describes this process by a pair of partial differential equations (PDEs). Thus they consider a mathematical model for the displacement of a solute through a medium which apart from a constant flow (advection) and a dispersion (diffusion) interacts with the medium by intermittent adsorption (the kinetics).

The paper is organized into 8 sections.

After an Introduction, in Section 2, the deterministic reactive transport model characterized by a pair of PDEs is introduced.

Then, in the next section, the behaviour of a single particle in the solute is described; and, in Section 4, the probability generating functions of the Markov binomial distribution which was described in the previous section is calculated.

Based on these, in Section 5, the convergence of the simple discrete-time stochastic model by letting the time step go to 0 is discussed.

In Section 6, it is shown that, in the case of instantaneous injection of the solute, the partial probability densities of the free and adsorbed parts of the solute do satisfy the PDEs defined in Section 2.

The means and variances of the established stochastic reactive transport model are computed in Section 7. In this sense, the authors mention that they obtain a correct formula for the free case in Proposition 7.2 (making reference to some results obtained previously by other authors).

The probability density function of the proposed stochastic reactive transport model is studied in Section 8. Remarks regarding to previous results obtained by other authors are made, too.

The paper finishes with some conclusions. – A good paper and a very instructive one.

The paper is organized into 8 sections.

After an Introduction, in Section 2, the deterministic reactive transport model characterized by a pair of PDEs is introduced.

Then, in the next section, the behaviour of a single particle in the solute is described; and, in Section 4, the probability generating functions of the Markov binomial distribution which was described in the previous section is calculated.

Based on these, in Section 5, the convergence of the simple discrete-time stochastic model by letting the time step go to 0 is discussed.

In Section 6, it is shown that, in the case of instantaneous injection of the solute, the partial probability densities of the free and adsorbed parts of the solute do satisfy the PDEs defined in Section 2.

The means and variances of the established stochastic reactive transport model are computed in Section 7. In this sense, the authors mention that they obtain a correct formula for the free case in Proposition 7.2 (making reference to some results obtained previously by other authors).

The probability density function of the proposed stochastic reactive transport model is studied in Section 8. Remarks regarding to previous results obtained by other authors are made, too.

The paper finishes with some conclusions. – A good paper and a very instructive one.

Reviewer: Gabriel V. Orman (Braşov)

### MSC:

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

### Keywords:

Markov binomial distribution; reactive transport; kinetic adsorption; solute transport; multimodality; double peak
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\textit{M. Dekking} and \textit{D. Kong}, Adv. Appl. Probab. 44, No. 3, 874--885 (2012; Zbl 1264.60053)

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