Continuous solutions of iterative equations of infinite order

Abstract

Given a probability space $(\Omega,\mathcal{A}, P)$ and a complete separable metric space $X$, we consider continuous and bounded solutions $\varphi: X \to \mathbb{R}$ of the equations $\varphi(x) = \int_{\Omega} \varphi(f(x,\omega))P(d\omega)$ and $\varphi(x) = 1-\int_{\Omega} \varphi(f(x,\omega))P(d\omega)$, assuming that the given function $f:X \times \Omega \to X$ is controlled by a random variable $L: \Omega \to (0,\infty)$ with $-\infty \lt \int_{\Omega} \log L(\omega)P(d\omega) \lt 0$. An application to a refinement type equation is also presented.