Have there been papers dealing with random walks in $\mathbb{R}^n$ that are not on the lattice $\mathbb{Z}^n$? Instead of walking in one of the directions possible in $\mathbb{Z}^n$ with probability $1/2^n$, one would walk in the direction corresponding to points on $S^{n1}$ with uniform probability. In the case of the lattice, the walk returns to the origin with probability $1$ for $n=1,2$ only. Do you know what happens in the continous case? Thanks.
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