Study of the corrector of the eigenvalue of a transport operator. (English) Zbl 0609.45002

Sentis, Rémi

SIAM J. Math. Anal. 16, 151-166 (1985).

In 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.

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.)

Keywords:

eigenvalue of a transport operator; discrete Markov processes; asymptotic expansion; largest eigenvalue; diffusion operator