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.