Convergence in total variation distance for (in)homogeneous Markov processes

In this paper, we study the rate of convergence in total variation distance for time continuous Markov processes, by using some Iψ and Iψ,t-inequalities. For homogeneous reversible process, we use some homogeneous inequalities, including the Poincaré and relative entropy inequalities. For the time-inhomogeneous diffusion process, we use some inhomogeneous inequalities, including the time-dependent Poincaré and Log-Sobolev inequalities. This extends some results for the time-homogeneous diffusion processes in Cattiaux and Guillin (2009).