  According to the Wiener-Hopf factorization, the characteristic function
$\varphi$ of any probability distribution $\mu$ on $\mathbb{R}$ can be
decomposed in a unique way as
\[1-s\varphi(t)=[1-\chi_-(s,it)][1-\chi_+(s,it)]\,,\;\;\;|s|\le1,\,t\in\mathbb{R}\,,\]
where $\chi_-(e^{iu},it)$ and $\chi_+(e^{iu},it)$ are the characteristic
functions of possibly defective distributions in
$\mathbb{Z}_+\times(-\infty,0)$ and $\mathbb{Z}_+\times[0,\infty)$,
respectively.
  We prove that $\mu$ can be characterized by the sole data of the upward
factor $\chi_+(s,it)$, $s\in[0,1)$, $t\in\mathbb{R}$ in many cases including
the cases where:
  1) $\mu$ has some exponential moments;
  2) the function $t\mapsto\mu(t,\infty)$ is completely monotone on
$(0,\infty)$;
  3) the density of $\mu$ on $[0,\infty)$ admits an analytic continuation on
$\mathbb{R}$.
  We conjecture that any probability distribution is actually characterized by
its upward factor. This conjecture is equivalent to the following: {\it Any
probability measure $\mu$ on $\mathbb{R}$ whose support is not included in
$(-\infty,0)$ is determined by its convolution powers $\mu^{*n}$, $n\ge1$
restricted to $[0,\infty)$}. We show that in many instances, the sole knowledge
of $\mu$ and $\mu^{*2}$ restricted to $[0,\infty)$ is actually sufficient to
determine $\mu$. Then we investigate the analogous problem in the framework of
infinitely divisible distributions.
