Study of the Corrector of the Eigenvalue of a Transport Operator
Rémi Sentis · SIAM Journal on Mathematical Analysis · 1985
We consider the transport operator\[A^e = \frac{{ - 1}}{\varepsilon }\sum_i {v_i } \frac{\partial }{{\partial x_i }} + \frac{1}{{\varepsilon ^2 }}Q\] 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. We show that its largest eigenvalue $\omega _\varepsilon $ converges (when $\varepsilon \to 0$) to the largest eigenvalue $\omega $ of a diffusion operator A on $L^2 (\Omega)$ and we calculate the limit of $({1 / \varepsilon })(\omega _\varepsilon - \omega )$.