For a talk I am going to prove Pólya's theorem, which states that a simple random walk on $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\ge 3$.
I want to show how this relates to Brownian motion, since a similar result holds. Is there an easy way that Pólya's theorem implies transience and recurrence in different dimensions? Brownian motion is often described as the limit of random walks but I am having trouble finding a reference that spells out the correspondence. For example, I looked at the Donsker’s invariance principle but this seems to only give you a Brownian motion defined for finite time.