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Yes, it is possible to extend the state space with respect to which $Y$ is a Feller process. Then, $X$ will be a dense open subset of the extension $\hat X$. Furthermore, for any initial distribution …

This is just an application of Ito's formula. Normally, Brownian motion $W(t)$ is only defined for nonnegative times $t$, so I'm assuming that $W(t)=0$ for $t\le0$, although it doesn't really matter m …

Having thought about this question some more, including making some plots of the trajectories of points under iterated applications of $f$ (see Gottfried's circular trajectories), it is possible to de …

Showing that the two definitions agree almost everywhere is easy! Using the truncated transform $$ \mathcal{H}\_\epsilon\mu(x)=\frac1\pi\int_{\lvert y-x\rvert > \epsilon}\frac{d\mu(y)}{x-y} $$ then, b …

No, convergence does not hold for all irrational $x$. I posted a full answer to the question at math.stackexchange and I'll summarize the result here. There are uncountably many values of $x$ for whi …

No. If the probability space contains atoms then you can easily construct sub-$\sigma$-algebras which are not independent to any nontrivial events. However, we can still construct examples, even for …

I learned the proof from this paper, Uniformization of Riemann Surfaces. They actually provide three methods of proof, and I found the first of these the easiest to follow. Its not too difficult to go …