Homogeneous conservative Wiener–Hopf equation

The existence of a $P^*$-solution of the homogeneous generalized Wiener–Hopf equation
$$ S(x)=\int_{-\infty}^xS(x-y)\,F(dy),\qquad x\geqslant0, $$
is proved, where $F$ is a probability distribution of recurrent type in $\mathbb R$. Asymptotic properties of this solution are established.
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