4 Occupations and Green’s functions of random walks

4.1 Regularized occupation

Regularized occupation of a walk

Definition 4.1

✓

Let \(\mathfrak {w}\colon \mathbb {N}\to \mathbb {Z}^d\) be a walk and let \(r \ge 0\). Then the \(r\)-regularized occupation of \(\mathfrak {w}\) at \(x \in \mathbb {Z}^d\) is

\begin{align} \label{eq: reg walk occupation} L^{\mathfrak {w}}_r(x) = \sum _{t \in \mathbb {N}} r^t \, \mathsf{1}_{\left\{ \mathfrak {w}(t) = x \right\} } . \end{align}

Lemma 4.2

✓

The \(r\)-regularized occupation \(L^{\mathfrak {w}}_r(x)\) of a walk \(\mathfrak {w}\) is an increasing function of \(r\).

Proof ▶

Each term in the defining sum 1 is increasing.

Lemma 4.3

✓

The \(r\)-regularized occupation \(L^{\mathfrak {w}}_r(x)\) of a walk \(\mathfrak {w}\) at any point \(x \in \mathbb {Z}^d\) satisfies \(L^{\mathfrak {w}}_r(x) \le \frac{1}{1-r}\).

Proof ▶

Calculate, using the definition 1:

\begin{align*} L^{\mathfrak {w}}_r(x) = \sum _{t \in \mathbb {N}} r^t \, \mathsf{1}_{\left\{ \mathfrak {w}(t) = x \right\} } \le \sum _{t \in \mathbb {N}} r^t = \frac{1}{1-r}. \end{align*}

Lemma 4.4

✓

The sum over all points \(x \in \mathbb {Z}^d\) of the \(r\)-regularized occupations \(L^{\mathfrak {w}}_r(x)\) of a walk \(\mathfrak {w}\colon \mathbb {N}\to \mathbb {Z}^d\) is

\begin{align*} \sum _{x \in \mathbb {Z}^d} L^{\mathfrak {w}}_r(x) = \sum _{t \in \mathbb {N}} r^t . \end{align*}

(Both sides above are infinite if \(r \ge 1\), but the equality nevertheless holds in \([0,+\infty ]\).)

Proof ▶

Unravel the definitions and interchange the two summations by Fubini’s theorem:

\begin{align*} \sum _{x \in \mathbb {Z}^d} L^{\mathfrak {w}}_r(x) = & \; \sum _{x \in \mathbb {Z}^d} \sum _{t \in \mathbb {N}} r^t \, \mathsf{1}_{\left\{ \mathfrak {w}(t) = x \right\} } \\ = & \; \sum _{t \in \mathbb {N}} \underbrace{\sum _{x \in \mathbb {Z}^d} r^t \, \mathsf{1}_{\left\{ \mathfrak {w}(t) = x \right\} }}_{=r^t}. \end{align*}

Here we noticed in the last expression that since at time \(t\) the walk is at exactly one point, there is exactly one value of \(x\) which contributes to the sum over \(x\), and this contribution is \(r^t\).

Regularized occupation of a random walk

Definition 4.5

✓

Let \(X\) be a random walk on \(\mathbb {Z}^d\) and let \(r \ge 0\). Then the \(r\)-regularized occupation of \(X\) at \(x \in \mathbb {Z}^d\) is

\begin{align} \label{eq: reg occupation} L_r(x) = L^{X}_r(x) = \sum _{t \in \mathbb {N}} r^t \, \mathsf{1}_{\left\{ X(t) = x \right\} } . \end{align}

Lemma 4.6

✓

The regularized occupation \(L_r(x)\) of a random walk \(RW\) is a \([0,+\infty ]\)-valued random variable.

Proof ▶

…

Lemma 4.7

✓

The \(r\)-regularized occupation \(L_r(x)\) of a random walk \(X\) is increasing in \(r\).

Proof ▶

This follows from Lemma 4.2.

Lemma 4.8

✓

The \(r\)-regularized occupation \(L_r(x)\) of a random walk \(X\) at any point \(x \in \mathbb {Z}^d\) satisfies \(L_r(x) \le \frac{1}{1-r}\).

Proof ▶

Follows from Lemma 4.3.

Lemma 4.9

✓

If \((r_n)_{n \in \mathbb {N}}\) is an increasing sequence with limit \(r = \lim _{n \to \infty } r_n\), then for any \(x \in \mathbb {Z}^d\) the random variables \(L_{r_n}(x)\) have limit \(L_{r}(x) = \lim _{n \to \infty } L_{r_n}(x)\).

Proof ▶

This uses a monotone convergence theorem: each term \(r_n^t \, \mathsf{1}_{\left\{ X(t) = x \right\} }\) of the infinite sum 2 is increasing and has limit \(r^t \, \mathsf{1}_{\left\{ X(t) = x \right\} }\) (by continuity of the \(t\)-power function \(\rho \mapsto \rho ^t\)), so the sum is increasing and the limit of the sum is the sum of the limits.

Lemma 4.10

✓

The sum over all points \(x \in \mathbb {Z}^d\) of the \(r\)-regularized occupations \(L_r(x)\) of a random walk \(X\) is

\begin{align*} \sum _{x \in \mathbb {Z}^d} L_r(x) = \sum _{t \in \mathbb {N}} r^t . \end{align*}

(Both sides above are infinite if \(r \ge 1\), but the equality nevertheless holds in \([0,+\infty ]\).)

Proof ▶

This is a random walk version of Lemma 4.4, but the right hand side expression does not depend on the walk, so there is no randomness involved (the right hand side is a constant random variable).

Lemma 4.11

✓

If \(r{\lt}1\), then the infinite sum in Lemma 4.10 is convergent in \(\mathbb {R}\), and the equality

\begin{align*} \sum _{x \in \mathbb {Z}^d} L_r(x) = \frac{1}{1-r} \end{align*}

holds in \(\mathbb {R}\).

Proof ▶

Apply Lemma 4.10 and the sum of a geometric series.

Lemma 4.12

✓

If \(r{\lt}1\), then the sum over points of the expected absolute values \(|L_r(x)|\) of the regularized occupation of a random walk \(X\) has the upper bound

\begin{align*} \sum _{x \in \mathbb {Z}^d} \mathsf{E}\Big[ \big| L_r(x) \big| \Big] \le \frac{1}{1-r} \end{align*}

Proof ▶

Interchange the expected value and the sum by Fubini’s theorem:

\begin{align*} \sum _{x \in \mathbb {Z}^d} \mathsf{E}\Big[ \big| L_r(x) \big| \Big] = \sum _{x \in \mathbb {Z}^d} \mathsf{E}\Big[ \big| L_r(x) \big| \Big] . \end{align*}

Then notice that \(|L_r(x)| = L_r(x)\), and apply Lemma 4.11.

4.2 Green’s function

Definition 4.13

✓

Let \(X\) be a random walk on \(\mathbb {Z}^d\) and let \(0 \le r {\lt} 1\). Then the \(r\)-regularized Green’s function of \(X\) is the function \(\mathrm{G}_{r} \colon \mathbb {Z}^d \to \mathbb {R}\) whose value at \(x \in \mathbb {Z}^d\) is the expected value of the regularized occupation \(L_r(x)\) at \(x\),

\begin{align} \label{eq: reg Green} \mathrm{G}_{r}(x) = \mathsf{E}\big[ L_r(x) \big] . \end{align}

Lemma 4.14

We have

\begin{align*} \sum _{x \in \mathbb {Z}^d} \mathrm{G}_{r}(x) = \mathsf{E}\Big[ \sum _{x \in \mathbb {Z}^d} L_r(x) \Big] . \end{align*}

Proof ▶

Interchange the expected value and the sum by Fubini’s theorem. The integrability hypotheses for the appropriate version of Fubini’s theorem follow from Lemma 4.12.

Lemma 4.15

We have

\begin{align*} \sum _{x \in \mathbb {Z}^d} \mathrm{G}_{r}(x) = \frac{1}{1-r} . \end{align*}

Proof ▶

First apply Lemma 4.14 to interchange the expected value and the sum over \(x\). Then apply Lemma 4.11. This yields

\begin{align*} \sum _{x \in \mathbb {Z}^d} \mathrm{G}_{r}(x) = \mathsf{E}\Big[ \sum _{x \in \mathbb {Z}^d} L_r(x) \Big] = \mathsf{E}\Big[ \frac{1}{1-r} \Big] = \frac{1}{1-r} , \end{align*}

where the last equality is just taking the expected value of a constant.

4.3 Expected occupation from the Green’s function

Lemma 4.16

Let \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) be a random walk starting from the origin \(\vec{0} \in \mathbb {Z}^d\). Denote by \(L = \# \left\{ t \in \mathbb {N}\, \Big| \, X(t) = \vec{0} \right\} \) the number of times the random walk is at the origin. Then we have

\begin{align*} \mathrm{G}_{r}(\vec{0}) \nearrow \mathsf{E}[L] \qquad \text{ as } r \nearrow 1 . \end{align*}

Proof ▶

Recall that \(\mathrm{G}_{r}(\vec{0}) = \mathsf{E}\big[ L_r(\vec{0}) \big]\) by definition, and observe that \(L = L_1(\vec{0})\). Therefore the statement is equivalent to \(\mathrm{G}_{r}(\vec{0}) \nearrow \mathrm{G}_{1} (\vec{0})\) as \(r \nearrow 1\). For this, it suffices to prove that whenever \((r_n)_{n \in \mathbb {N}}\) is an increasing sequence with limit \(1 = \lim _{n \to \infty } r_n\), then \(\mathrm{G}_{r_n}(\vec{0}) \nearrow \mathrm{G}_{1}(\vec{0})\) as \(n \to \infty \). Lemma 4.9 shows exactly that.