2 Random walks
2.1 Random walks on the \(d\)-dimensional integer grid
The \(d\)-dimensional integer grid is \(\mathbb {Z}^d\).
A walk on the grid \(\mathbb {Z}^d\) is a function \(\mathfrak {w}\colon \mathbb {N}\to \mathbb {Z}^d\), denoted \(t \mapsto \mathfrak {w}(t)\)). We construct walks from their sequences of steps as follows:
A sequence \((u_s)_{s \in \mathbb {N}}\) of steps in \(\mathbb {Z}^d\) determines a walk \(\mathfrak {w}\colon \mathbb {N}\to \mathbb {Z}^d\) by
A random walk is constructed from a sequence of random steps.
A sequence \((\xi _s)_{s \in \mathbb {N}}\) of \(\mathbb {Z}^d\)-valued random variables (on some a probability space) determines a random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) by
The position \(X(t)\) of a random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) at any time \(t \in \mathbb {N}\) is a \(\mathbb {Z}^d\)-valued random variable.
We must prove that \(X(t) \colon \Omega \to \mathbb {Z}^d\) is measurable. Since each of the steps \(\xi _s \colon \Omega \to \mathbb {Z}^d\) is measurable by assumption and the position of the random walk is defined in 1 by summing the first \(t\) steps, the measurability follows by induction on \(t\) using the fact that sums of measurable \(\mathbb {Z}^d\)-valued functions are measurable.
A random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) said to be time-homogeneous Markovian if its steps \(\big(X(t+1) - X(t)\big)_{t \in \mathbb {N}}\) are independent and identically distributed.
A random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) is simple if it is time-homogeneous Markovian and its steps are uniformly distributed on nearest neighbors on the grid: