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**Random motions in inhomogeneous media.**
*(Russian, English)*
Zbl 1199.60349

Teor. Jmovirn. Mat. Stat. 76, 125-137 (2007); translation in Theory Probab. Math. Stat. 76, 141-153 (2008).

The author deals with a transport equation of the following form

\[ \frac{\partial}{\partial t}p(x,v,t)+v\cdot\nabla p(x,v,t)=-\mu p(x,v,t) +\mu\int_VT(v,v')p(x,v',t)dv', \]

where \(p(x,v,t)\) denotes the density of particles at a spatial position \(x\in\mathbb R^n\) moving with velocity \(v\in V \subset\mathbb R^n\) at time \(t\geq0\) and \(\mu\) is a constant turning rate. The turning kernel \(T(v,v')\) is the probability that the velocity jumps from \(v'\) to \(v\) provided the jump occurs.

This general model applies in various fields from thermal conductivity and photon transport in a highly scattering medium [D. D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, No. 1, 41–73 (1989); Addendum 62, 375 (1990; Zbl 1129.80300); G. H. Weiss Physica A 311, No. 3-4, 381–410 (2002; Zbl 0996.35040)] to description of the motion of biological organisms [K. P. Hadeler, in: Mathematics inspired by biology, Berlin: Springer. Lect. Notes Math. 1714, 95–150 (1999; Zbl 1002.92506); H. G. Othmer, S. R. Dunbar and W. Alt, J. Math. Biol. 26, No. 3, 263–298 (1988; Zbl 0713.92018); T. Hillen and H. G. Othmer [SIAM J. Appl. Math. 61, No. 3, 751–775 (2000; Zbl 1002.35120)]. The solution of the full transport equation can only be found numerically (see Weiss, loc.cit.). Thus it is natural to seek stochastic models and the corresponding differential equations that are simpler than the general equation. In the one-dimensional case, this equation is equivalent to the so-called telegraph equation, which is a widespread research project beginning from the middle of the 19th century (see the survey by Joseph and Preziosi, loc. cit.).

However, the reduction of the process with jump velocities to the telegraph equation is possible in the one-dimensional case only. Even if only four directions are allowed in the plane, the system does not reduce to the telegraph equation in any scaling. Nevertheless the equations can be solved in quadratures in the latter case. An alternative approach to the transport equation leads to the telegraph equation for all dimensions (see Weiss, loc. cit.).

These two approaches do not coincide. In the article under review the first approach is exploited. The idea of a rectifying diffeomorphism is developed to construct explicit solutions for inhomogeneous media. The results are illustrated by examples of physical and biological processes. In particular, the diffusions in the medium with circular or cellular structure can be given in terms of velocity-jump processes with varying velocity.

\[ \frac{\partial}{\partial t}p(x,v,t)+v\cdot\nabla p(x,v,t)=-\mu p(x,v,t) +\mu\int_VT(v,v')p(x,v',t)dv', \]

where \(p(x,v,t)\) denotes the density of particles at a spatial position \(x\in\mathbb R^n\) moving with velocity \(v\in V \subset\mathbb R^n\) at time \(t\geq0\) and \(\mu\) is a constant turning rate. The turning kernel \(T(v,v')\) is the probability that the velocity jumps from \(v'\) to \(v\) provided the jump occurs.

This general model applies in various fields from thermal conductivity and photon transport in a highly scattering medium [D. D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, No. 1, 41–73 (1989); Addendum 62, 375 (1990; Zbl 1129.80300); G. H. Weiss Physica A 311, No. 3-4, 381–410 (2002; Zbl 0996.35040)] to description of the motion of biological organisms [K. P. Hadeler, in: Mathematics inspired by biology, Berlin: Springer. Lect. Notes Math. 1714, 95–150 (1999; Zbl 1002.92506); H. G. Othmer, S. R. Dunbar and W. Alt, J. Math. Biol. 26, No. 3, 263–298 (1988; Zbl 0713.92018); T. Hillen and H. G. Othmer [SIAM J. Appl. Math. 61, No. 3, 751–775 (2000; Zbl 1002.35120)]. The solution of the full transport equation can only be found numerically (see Weiss, loc.cit.). Thus it is natural to seek stochastic models and the corresponding differential equations that are simpler than the general equation. In the one-dimensional case, this equation is equivalent to the so-called telegraph equation, which is a widespread research project beginning from the middle of the 19th century (see the survey by Joseph and Preziosi, loc. cit.).

However, the reduction of the process with jump velocities to the telegraph equation is possible in the one-dimensional case only. Even if only four directions are allowed in the plane, the system does not reduce to the telegraph equation in any scaling. Nevertheless the equations can be solved in quadratures in the latter case. An alternative approach to the transport equation leads to the telegraph equation for all dimensions (see Weiss, loc. cit.).

These two approaches do not coincide. In the article under review the first approach is exploited. The idea of a rectifying diffeomorphism is developed to construct explicit solutions for inhomogeneous media. The results are illustrated by examples of physical and biological processes. In particular, the diffusions in the medium with circular or cellular structure can be given in terms of velocity-jump processes with varying velocity.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

60K99 | Special processes |

62G30 | Order statistics; empirical distribution functions |

60C05 | Combinatorial probability |

35L25 | Higher-order hyperbolic equations |