Differential entropy time-dependent density function

Question

The typical definition of Differential Entropy is given as:

$h(X) = -\int f(X) \log f(X) dX$.

Can this be extended to time-dependent probability density functions, $f(X,t)$? Perhaps leading to a time-dependent entropy?

Thanks.

Answer 1

As the comment says, your current form is true for a single time point. So it is true for any arbitrary time point $t$ to write: $$ h(X,t) = -\int f(X,t)\, \log f(X,t) \, dX $$