0
$\begingroup$

Let $$S = \{(x_1,x_2,\ldots,x_d) \in \mathbb{Z}^d : x_i > 0, i=1,\ldots,d \}$$ and consider a simple and symmetric random walk starting in a point $x_0 \in S$.

I wish to know more properties about the probability $P(x_0)$ that this random walk always stays in $S$. From Polya's recurrence theorem, I know that $P(x_0) = 0$ for $d=1$ and for $d=2$.

What can be said for $d \ge 3$?

  1. In particular, is there an explicit expression for $P(x_0)$?
  2. If not, it is true that $P(x_0) > 0$ for all $x_0 \in S$ ?
$\endgroup$

1 Answer 1

1
$\begingroup$

There is at least one direction in which the walk takes infinitely many steps. (In fact it almost surely takes infinitely many steps in all $d$ directions, but we don’t need that.) The movement in this direction is a simple symmetric random walk in one dimension. The probability that it stays on the same side of the origin forever is $0$. Thus the probability that the $d$-dimensional walk stays in the same orthant forever is also $0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.