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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$$E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}.$$

And here.

It follows that for $x$, $\theta \in \mathbb{R}$ and $y > 0$,$$E^{(x, y)} e^{i\theta X_\tau} = e^{i\theta x - |\theta|y}.$$

My question is, how do we use Fourier inversion to calculate the density of the random variable $X_\tau$, to conclude that the Poisson kernel is indeed the Cauchy distribution?

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