The move is uniformly random in four directions (u/d/l/r), at 1 unit. What is the expected number of moves? What about from (0,0) to (2, 2)?
I have also an easier version: what is the expected number of moves from 0 to 2 on number axis, if the move is uniformly random in both directions?
I have tried to create equations but it seems I need an infinite number of equations.
The expected return time to the origin is infinite already for a one-dimensional random walk. Since going from $(0,0)$ to $(1,1)$ includes going from a nearest neighbour of $(1,1)$ (say, $(0,1)$) to $(1,1)$ and the horizontal component of this walk is a one-dimensional walk that tries to return to the origin from its nearest neighbour, the expected time to get from $(0,0)$ to $(1,1)$ is also infinite.