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 ε (the mean free path), tends to zero. We show that, for arbitrary initial data u₀(x, v), the solution converges in norm topology for each t > 0, to the solution of a diffusion equation with initial data u_D⁰(x) = ∫ u₀(x, v)dv. The proof relies on Fourier analysis which diagonalizes the transport operator, a Dunford functional calculus and the analysis of the behaviour of the transport spectrum as ε tends to zero.