A Random Fixed Point Theorem and the Random Graph Transformation

In the first part of this article we formulate a random fixed point theorem for random dynamical systems where a contraction condition is formulated as an expectation of a particular expression. In the following we are going to formulate a random graph transformation. This transformation defines a random dynamical system. Under certain conditions this random dynamical system has a random fixed point which can be found by the mentioned random fixed point theorem. In one application we show that random dynamical systems given by particular differential equations have a random unstable invariant manifold.