Could someone helpme with this problem?
First, consider the transition kernel in $\mathbb{R^2}\times B(\mathbb{R})$ given by $K(x,A)=U_{S^1}(A-x)$. We can than define an other kernel in $\mathbb{R^4}\times B(\mathbb{R^2})$ by $K_2((x_1,x_2),A)=K(x_2,A)$. My goal is to find a probability $\mu$ in $\mathbb{R^2}$ that is given by $\mu=\delta_0K_2$, where $\delta_0$ is the dirac mass centered at the origin.
I was thinking that one way of trying this is to prove that $\mu$ is invariant under $O(2)$ action and to calculate the probability of balls centered at the origin.
Does someone has some idea to solve the problem?