Stochastic operators,information, and entropy

For a stochastic operatorU on andL ₁-space, i.eU is linear, positive, and norm preserving on the positive cone ofL ₁, it is shown thatU decreases relative information between two nonnegativeL ₁-functions. Furthermore it is shown that the following properties ofU are closely related:U is energy decreasing (energy preserving),U isH-decreasing, whereH is Boltzmann'sH-functional, and the Maxwell distributions are fixed points ofU.