Statistical Thermodynamics of General Minimal Diffusion Processes:   Constuction, Invariant Density, Reversibility and Entropy Production
  The solution to nonlinear Fokker-Planck equation is constructed in terms of the minimal Markov semigroup generated by the equation. The semigroup is obtained by a purely functional analytical method via Hille-Yosida theorem. The existence of the positive invariant measure with density is established and a weak form of Foguel alternative proven. We show the equivalence among self-adjoint of the elliptic operator, time-reversibility, and zero entropy production rate of the stationary diffusion process. A thermodynamic theory for diffusion processes emerges. 
Hong Qian, Min Qian, and Xiang Tang