 Normalized velocity-reset collision operator

Let $M\ge 0$ have velocity integral one, and $\rho (g)(x)=\int g(x,v)dv$. On phase-space $L^1$, the normalized relaxation collision operator is $B=P-I$, where $Pg=\rho (g)M$. It obeys $\Vert B\Vert \le 2$ because $\Vert Pg{\Vert }_1\le \Vert g{\Vert }_1$. It preserves mass at each spatial position. The corresponding linear Boltzmann equation combines this bounded collision operator with the free-transport semigroup.