What is the proability to return to the origin, for a uniform random walk on the integer lattice in $\mathbb Z^3$, if we are restricted to $x \geq 0$? I.e. if we try to step into a negative x-coordinate we reflect and go $1$ step in positive $x$-direction.
And what if we are restritced to a quarter of $\mathbb Z^3$, i.e. $x \geq 0$, $y \geq 0$?
And $z \geq 0,y \geq 0, x \geq 0$?
We start at origin and move $1$ unit in $x$, $y$, $z$, $-x$, $-y$ or $-z$ direction with probability $\frac16$ at each step.