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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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You can find the relevant calculation here.

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  • $\begingroup$ thank you very much for the reference. This seems to come from the fact that for a test function f, the following inequality L log(f) >= Lf/f is true. I can see that in the case of diffusions (for example), with generator of the form Lf = mu.f' + sigma^2.f'' but I still do not see why this should be true for a more general generator L. Is it ? $\endgroup$ – Alekk Apr 9 '10 at 15:06

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