We know that Brownian motion describes any (non-deterministic) Lévy process with continuous sample paths.
The above statement is true in Euclidean space. My question is: does it stand for other topological spaces too, and in particular, is it valid over the Lie group $\mathbb{U}(n)$, that is, the group of $n \times n$ unitary matrices?
My intuition is "yes", since such a process can be induced by a Brownian motion over its Lie algebra, where the above mentioned statement surely stands, but on the other hand I know that in a topology such as $\mathbb{U}(n)$ some properties of Brownian motion, like the normal distribution of the increments, are not valid anymore.
Is there a reference or a proof confirming or rejecting the statement for Brownian motion over a Lie group, or even more generally, over any topological space?