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 $S^{n-1}$ 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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3$\begingroup$ In the real case you need to change the definition of recurrence: there's 0 probability of ever hitting any specific point, so you look instead at recurrence to a neighbourhood (i.e. a random walk is recurrent if for each initial point $x$ and for each $\epsilon > 0$, the walk almost surely returns to an $\epsilon$-neighbourhood of $x$. With this definition, the results are the same as in the discrete case: the random walk is recurrent in 1 and 2 dimensions; and transient in 3 dimensions or higher. $\endgroup$– Anthony QuasCommented Feb 17, 2012 at 6:44
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$\begingroup$ That was very helpful. Thank you very much. $\endgroup$– alexeiCommented Feb 17, 2012 at 7:55
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1$\begingroup$ A related result is Rayleigh's theorem: start at the origin in the plane and make $n$ steps of length one, with directions chosen independently and uniformly at each step. Then the probability of ending up at distance less than one from the origin is $1/(n+1)$. For an elegant recent proof by Bernardi, see math.mit.edu/~bernardi/publications/Rayleigh.pdf. $\endgroup$– Richard StanleyCommented Feb 17, 2012 at 14:18
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$\begingroup$ Look for some sources ... en.wikipedia.org/wiki/Random_walk $\endgroup$– Gerald EdgarCommented Feb 17, 2012 at 14:39
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