Study of the corrector of the eigenvalue of a transport operator.

*(English)*Zbl 0609.45002In his former paper [SIAM J. Appl. Math. 39, 134-141 (1980; Zbl 0441.60100)] the author considered a transport process such that the mean free path between two successive jumps is of order \(\epsilon\) and such that the structure of the medium is periodic with period \(\delta =\epsilon^ k\) \((k<1)\) in each direction. It is proved that this process converges weakly (when \(\epsilon\to 0)\) to a diffusion, while various other results referred to the transport equation were established in his Ph.D. Thesis ”Analyse asymptotique d’équation de transport avec structure périodique” (Université Paris IX, 1981).

In this paper the author considers the transport operator \[ A^{\epsilon}=-1/\epsilon \sum_{i}v_ i(\partial /\partial x_ i)+(1/\epsilon^ 2)Q \] on the space \(L^ 2(\Omega \times V)\) where \(\Omega\) is a bounded open set of \(R^ N\), V a compact set of \(R^ N\) and Q a Markovian generator. The goal of this paper is to give an asymptotic expansion of its largest eigenvalue \(\omega_{\epsilon}\) with respect to \(\epsilon\). It is proved that \(\omega_{\epsilon}\) converges (when \(\epsilon\to 0)\) to the largest eigenvalue \(\omega\) of a diffusion operator A on \(L^ 2(\Omega)\). The limit of \((1/\epsilon)(\omega_{\epsilon}-\omega)\) is also calculated. There are also 5 remarks, from which, for instance, the first one asserts that for \(\epsilon\) small enough, the dimension of the eigenspace of \(A^{\epsilon}\) related to \(\omega_{\epsilon}\) is 1.

In this paper the author considers the transport operator \[ A^{\epsilon}=-1/\epsilon \sum_{i}v_ i(\partial /\partial x_ i)+(1/\epsilon^ 2)Q \] on the space \(L^ 2(\Omega \times V)\) where \(\Omega\) is a bounded open set of \(R^ N\), V a compact set of \(R^ N\) and Q a Markovian generator. The goal of this paper is to give an asymptotic expansion of its largest eigenvalue \(\omega_{\epsilon}\) with respect to \(\epsilon\). It is proved that \(\omega_{\epsilon}\) converges (when \(\epsilon\to 0)\) to the largest eigenvalue \(\omega\) of a diffusion operator A on \(L^ 2(\Omega)\). The limit of \((1/\epsilon)(\omega_{\epsilon}-\omega)\) is also calculated. There are also 5 remarks, from which, for instance, the first one asserts that for \(\epsilon\) small enough, the dimension of the eigenspace of \(A^{\epsilon}\) related to \(\omega_{\epsilon}\) is 1.

Reviewer: D.Mangeron

##### MSC:

45C05 | Eigenvalue problems for integral equations |

45K05 | Integro-partial differential equations |

47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |

82C70 | Transport processes in time-dependent statistical mechanics |

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