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### Poisson kernel, expectation, an absolute value comes in

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...

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### Poisson kernel is the Cauchy distribution, reference?

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?

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### Predictable quadratic Variation <.> has same intervals of constancy as the process

From
Revuz and Yor - Continuous Martingales and Brownian Motion 1999
Chapter IV Proposition 1.13
it is proven, that for a continuous local martingale $M_t$ the intervals of ...