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