On the uniqueness of the solution to the Wiener-Hopf equation with probability kernel (Q2058310)

The inhomogeneous generalised Wiener-Hopf equation is \[ z(x) = \int_{-\infty}^x z(x-y)\, F(dy) + f(x), \qquad x\geq 0, \] where \(z\) is the unknown function, \(F\) is a given probability distribution and \(f\) is a known function. A distribution \(F\) is called arithmetic if it is concentrated at a set \(\lambda \mathbb{Z}\) (for some non-zero \(\lambda\in\mathbb{R}\)), otherwise it is called nonarithmetic. For a nonarithmetic \(F\) with a finite positive mean \(\mu=\int_{\mathbb{R}} x \,F(dx)\) the uniqueness of the solution is proven. The method is based on an elaborated von Neumann series expansion \((I-\mathcal{F})^{-1}=\sum_0^{\infty} \mathcal{F}^n\) for convolutions. An analogous result for a discrete case with arithmetic \(F\) is discussed as well.