I would like to parameterize all the possible modifications to a probability density function. Is there a representation theory for this? Something along the lines of, these are all the linear operators $L$ that act on a pdf $f(x)$ such that $(1 + \epsilon L)f$ (for small $\epsilon$) is positive everywhere and still integrates to $1$.
I guess it must be some kind of infinite-dimensional symmetry group but hopefully there's some relatively convenient way to parameterize all such transformations.