On the singular limit of a model transport semigroup

We consider the diffusion limit of a model transport equation on the torus or the whole space, as a scaling parameter epsilon (the mean free path), te nds to zero. We show that, for arbitrary initial data u(0)(x, upsilon) the solution converges in norm topology for each t > 0, to the solution of a di ffusion equation with initial data u(D)(0)(x) = integral u(0)(x, upsilon) d upsilon. The proof relies on Fourier analysis which diagonalizes the trans port operator, a Dunford functional calculus and the analysis of the behavi our of the transport spectrum as epsilon tends to zero. Copyright (C) 2000 John Wiley & Sons, Ltd.