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

The characteristic function $\varphi$ of any probability distribution $\mu$ on $\mathbb{R}$ can be decomposed as $[1-\varphi(t)=[1-\kappa_+(s,t)]\cdot[1-\kappa_-(s,t)],s\in[0,1),,; t\in\mathbb{R}$ where $\kappa_+$ and $\kappa_-$ are respectively the upward and downward space-time Wiener-Hopf factors of $\mu$. The latter factors are defined by $[kappa_+(s,t)=e(s^{T_+}e^{itS_{T_+}});\mbox{and};\kappa_-(s,t)=e(s^{T_-}e^{itS_{T_-}})]$ where $(S_n)$ is a random walk with step distribution $\mu$, starting at 0 and $T_+,T_-$ are the first passage times above and bellow 0 by $(S_n)$, that is $T_+=\inf\{n\ge1:S_n>0\}$ and $T_-=\inf\{n\ge1:S_n\le0\}$. We prove that $\mu$ can be characterized by the sole data of the upward factor $\kappa_+(s,t)$, $s\in[0,1)$, $t\in\mathbb{R}$ in many cases including the case where 1) $\mu$ has some positive exponential moments, 2) the function $t\mapsto\mu(t,\infty)$ is completely monotone on $\mathbb{R}_+$, 3) the density of $\mu$ in $\mathbb{R}_+$ satisfies some conditions of analycity. We conjecture that any probability distribution is 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 $\mathbb{R}_-$ is determined by its convolution iterations $\mu^{*n}$, $n\ge1$ restricted to $\mathbb{R}_+$}. In many instances, the sole knowlege of $\mu$ and $\mu^{*2}$ restricted to $\mathbb{R}_+$ is actually sufficient to determine $\mu$. This is a joint work with Ron Doney (Manchester University).