$X$ are $X'$ are two independent and identical distributed variables.
Given the distribution of $X-X'$, is the distribution of $X$ identifiable up to mean shift? If yes, how to compute it? If no, what can we know about $X$?
I can think of the equation
$$\forall c\in \mathbb{R}^+, F_{X-X'}(c) = \int_{-\infty}^{\infty}f_X(x')\int_{-\infty}^{x'+c}f_X(x) \ dxdx',$$
where $F$ and $f$ denotes CDF and PDF respectively. But I don't know how to further solve $f_X$ from $F_{X-X'}$. It seems a convolution operation. Are there explicit solution expressions?