Every -complex is homotopy equivalent to (the realization of) a simplicial complex.
For , let be a complex domain and let . Let be a complex Brownian motion starting from . Set .
Suppose that there exists a conformal isomorphism such that . Set and define for
Then and have the same distribution.
Let such that . Prove that .
Let be the symmetric random walk in –, starting at the origin. So, and the transition probabilities are if and otherwise.
Let be the orthogonal projections of the random walk onto the diagonals and .
- Prove that are independent symmetric random walks in such that if and only if .
- Prove that , as , where and . Conclude that . This is equivalent to the recurrence of the random walk.
Prove that the planar Brownian motion is neighbourhood recurrent, but not point recurrent. Conclude that the path of planar Brownian motion is dense in the plane.
Let be a sequence of entire functions such that, for every , there exists . Suppose that is continuous and . Let be an entire function such that . Prove that over compact sets if and only if , for every .