Take a Markov process $X_t$ on a Polish state-space $E$. Suppose the Markov process has invariant measure $\mu$ and is prepared with initial distribution $\mu_0\ll \mu$. De Bruijn's identity says that the relative entropy of the law of $X_t$ wrt the stationary measure $\mu$ is always decreasing, and the rate of decrease is given by the Fisher information $I$ \begin{equation*} \frac{d}{dt}H \left(\mu_t \| \mu \right) = - I \left(\mu_t, \mu\right) \leq 0 \end{equation*}
Do we have the similar property of $H \left( \mu \|\mu_t\right)$ also decreasing over time? And what can we say about $\frac{d}{dt}H \left( \mu \|\mu_t\right)$?
EDIT: Baez and Pollard show in equation (14) of their paper Relative Entropy in Biological Systems (2016) that $\frac{d}{dt}H \left( \mu \|\mu_t\right)\leq 0$ for any finite state-space Markov process. However, they do not discuss the general case.