On distributions determined by their upward, space-time Wiener-Hopf factor

According to the Wiener-Hopf factorization, the characteristic function $\phi$ of any probability distribution $\mu$ on $\mathbb{R}$ can be decomposed in a unique way as $$1-s\phi (t)=[1-{\chi }_{-}(s,it)][1-{\chi }_{+}(s,it)],\mathrm{\mid }s\mathrm{\mid }\le 1,t\in \mathbb{R},$$ where ${\chi }_{-}(e^{iu},it)$ and ${\chi }_{+}(e^{iu},it)$ are the characteristic functions of possibly defective distributions in ${\mathbb{Z}}_{+}\times (-\mathrm{\infty },0)$ and ${\mathbb{Z}}_{+}\times [0,\mathrm{\infty })$, 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 $t\mapsto \mu (t,\mathrm{\infty })$ is completely monotone on $(0,\mathrm{\infty })$;
3. the density of $\mu$ on $[0,\mathrm{\infty })$ 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 $(-\mathrm{\infty },0)$ is determined by its convolution powers ${\mu }^{\ast n}$, $n\ge 1$ restricted to $[0,\mathrm{\infty })$}. We show that in many instances, the sole knowledge of $\mu$ and ${\mu }^{\ast 2}$ restricted to $[0,\mathrm{\infty })$ is actually sufficient to determine $\mu$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.