Please tell me if I'm in the wrong place to ask this question. I thought this might be the best place to ask since my question is about the formal properties of some function.
If I have some dynamic PDF, $f(x,p;t)$, and some stationary PDF, $f_{eq}(x,p)$, I can write this as the sum \begin{equation} f(x,p;t) = f_{eq}(x,p) + \delta f(x,p;t) \end{equation} where $\delta f$ just quantifies the pointwise difference between the stationary and non-stationary PDF.
This kind of thing comes up in physics, for instance, when dealing with the Boltzmann equation. What I'm wondering is, what is the right way to talk about what's being done here? It seems formally strange to me that the object on the left is a pure PDF, and the objects on the right are a pure PDF and some other function that can't be a PDF since it can be negative. Although, it clearly obeys some normalization conditions. Does this kind of decomposition have a name? Has it been studied in other scenarios? I'm quite ignorant here and just looking for any sort of formal guidance in dealing with this kind of object. Any help is appreciated!