I wish to find all probability distributions $\mu$ on $\mathbb R^+$ whose Laplace transform:
$\varphi(z) = \int_{0}^{\infty} e^{-zx} \mu(dx), z \geqslant 0,$
solves the equation:
$\varphi(z) = \beta \varphi(z) g(z) + (1-\beta) \frac{\varphi(z+1)}{\varphi(1)},$
where $g$ again is the Laplace transform of some given distribution on $\mathbb R^+$
This equation characterizes the fixed point in distribution of some probabilistic model for the balance between selection and mutation. The difficulty comes from the non-local term (the fact that the equation mixes $z$ and $z+1$)
All comments are welcome !