pr.probability - Entropy of Markov processes

Consider a Markov process $X_t$ with generator $L$ and invariant distribution $pi$, whose distribution at time $t$ is given by $pi(t,dx)=phi(t,x) pi(dx)$ - in other word, $phi(t,x)$ is the density of $pi(t, dx)$ wiht respect to the invariant distribution $pi$. Define the (relative) entropy

$$S(t) = -int phi(t,x) ln phi(t,x) pi(dx) leq 0.$$

One can expect (Boltzman H-theorem) the entropy $S$ to increase over time, and eventually to converge to $0$.

question: what conditions should be imposed in order for such a result to be true ?

Fokker-Planck equation shows that for any test function $f$,

$$int f(x) partial_t phi(t,x) pi(dx) = int (Lf)(x) phi(t,x) pi(dx)$$
so that
$$S'(t) = -int L (ln circ phi)(x,t) phi(x,t) pi(dx),$$
but I still do not see why this quantity should be non-negative.

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