Collision operators as generators of Markov processes and their spectra

Abstract

The Markovian description of diffusion in velocity space involves a semigroup, which because of detailed balance is conveniently interpreted in a weightedL ²-space. The collision operatorC, defined by the corresponding generator, is positive semidefinite in this space. For a jump process and a continuous process we obtain the collision operators of the linear Boltzmann and Fokker-PIanck equations, respectively. If in the latter case the friction tensor has a nonvanishing limit asυ → ∞, the spectrum ofC is discrete. The Fourier-transformed transport operatorT _k=C+ik·v is studied as a holomorphic family of sectorial operators. In the stated Fokker-Planck example, the spectrum ofT _k remains discrete for arbitrary k.