# Restricted random walk in $\mathbb Z^3$

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.

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The reflections don't change the probability to return to the origin, since that probability is independent of the signs of the coordinates. Thus the probability to return to the origin is about $34\%$, as calculated here.