Homogeneous conservative Wiener-Hopf equation. (English. Russian original) Zbl 1143.45001

Sgibnev, M. S.

Sb. Math. 198, No. 9, 1341-1350 (2007); translation from Mat. Sb. 198, No. 9, 123-132 (2007).

A function \(S(x)\) satisfying the equation
\[ S(x)= \int^\infty_0 k(x-y)S(y)\,dy,\;x> 0,\quad k(x)= k(-x)\geq 0,\quad \int^\infty_{-\infty} k(x)\,dx= 1, \] is said to be its \(P^*\)-solution if it is nondecreasing, right-continuous, non-trivial and \(S(x)= 0\) for \(x< 0\).
Main result: The renewal function \(u_+(x)\) is a \(P^*\)-solution of the homogeneous generalized Wiener-Hopf equation
\[ S(x)= \int^x_{-\infty} S(x-y) F(dy),\quad x\geq 0, \] where \(F\) is a distribution of the recurrent type, with the condition \(u_+(0+)= 1\). Asymptotic properties of such solutions are also studied.

Reviewer: D. Przeworska-Rolewicz (Warszawa)

MSC:

45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
60G50	Sums of independent random variables; random walks

Keywords:

renewal function; random walk; symmetric probability distribution of recurrent type