Dependencies

A random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) is expectation recurrent if and only if \(\lim _{r \nearrow 1} \, I_r = +\infty \). In other words, \(X\) is expectation transient if and only if \(\lim _{r \nearrow 1} \, I_r \, {\lt} \, +\infty \).

Denote by \(L = \# \left\{ t \in \mathbb {N}\, \Big| \, X(t) = x_0 \right\} \) the number of times the random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) is at its starting point. The random walk \(X\) is said to be expectation recurrent if \(\mathsf{E}[ L ] \, = \, +\infty \) and expectation transient if \(\mathsf{E}[ L ] \; {\lt} \; +\infty \).

For any \(\delta {\gt} 0\), define the integrals

where \(B_\delta := \left\{ \theta \in \mathbb {R}^d \, \big| \, \| \theta \| {\lt} \delta \right\} \) is the ball of radius \(\delta \) centered at \(\vec{0} \in \mathbb {R}^d\).

Let \(0 \le r {\lt} 1\). The Fourier transform of the regularized Green’s function \(\mathrm{G}_{r} \, \colon \, \mathbb {Z}^d \to \mathbb {R}\) is the function

given by

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 \(x\),

The \(d\)-dimensional integer grid is \(\mathbb {Z}^d\).

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.

The random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) is said to be first return recurrent if \(\mathsf{P}\big[ X(t) = x_0 \text{ for some } t {\gt} 0 \big] \, = \, 1\) and first return transient if \(\mathsf{P}\big[ X(t) = x_0 \text{ for some } t {\gt} 0 \big] \; {\lt} \; 1\).

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 random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) is said to be recurrent if

and transient if

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

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:

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

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

For any \(x \in \mathbb {Z}^d\) and \(0 \le r {\lt} 1\), we have

For any \(x \in \mathbb {Z}^d\) and \(0 \le r {\lt} 1\), we have

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

If \(X\) is a time-homogeneous Markovian random walk with suitable non-degeneracy conditions on its step distribution (to be written down more precisely), then for any \(0 {\lt} \delta \le \pi \) the limit

exists and is finite (limit in \(\mathbb {R}\)).

For any \(0 {\lt} \delta \le \pi \) we can write

If \(X\) is a time-homogeneous Markovian random walk with suitable symmetricity and integrability conditions on its step distribution (to be written down more precisely), then there exists a \(\delta _0 {\gt} 0\) such that for any \(0 {\lt} \delta \le \delta _0\), the limit

is increasing and exists in \([0,+\infty ]\).

For a time-homogeneous random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) with step distribution \(p \colon \mathbb {Z}^d \to [0,1]\), the Fourier transform of the Green’s function is

A nice random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) is expectation recurrent if and only if for any small \(\delta {\gt}0\) \(\lim _{r \nearrow 1} \, K_r^{(\delta )} = +\infty \). In other words, \(X\) is expectation transient if and only if for some small \(\delta {\gt}0\) we have \(\lim _{r \nearrow 1} \, K_r^{(\delta )} \, {\lt} \, +\infty \).

A Markovian random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) is recurrent if and only if it is expectation recurrent.

A Markovian random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) is recurrent if and only if it is first return recurrent.

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

For \(0 \le r {\lt} 1\), the regularized Green’s function \(\mathrm{G}_{r} \colon \mathbb {Z}^d \to \mathbb {R}\) is an element of the Hilbert space \(\ell ^2(\mathbb {Z}^d, \mathbb {C})\) of complex-valued square-summable functions on \(\mathbb {Z}^d\).

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

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

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 \(L_{r}(x) = \lim _{n \to \infty } L_{r_n}(x)\).

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.

For the simple random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\), the Fourier transform of the Green’s function is

For the simple random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\), there exists a small \(\delta _0 {\gt} 0\) (something like \(\delta _0 = \frac{\pi }{8}\)) such that for all \(\theta \in B_{\delta _0} \setminus \left\{ 0 \right\} \) we have \(\Re \mathfrak {e}\big( \widehat{\mathrm{G}}_{r}(\theta ) \big) \uparrow \Re \mathfrak {e}\big( \widehat{\mathrm{G}}_{1}(\theta ) \big)\) as \(r \uparrow 1\) and

For the simple random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) with \(d {\gt} 2\), we have \(\lim _{r \nearrow 1} \, K_r^{(\delta )} \, {\lt} \, +\infty \) for \(\delta {\gt} 0\) small enough.

For the simple random walk \(X= \big(X(t)\big)_{t \in \mathbb {N}}\) on \(\mathbb {Z}^d\) with \(d \le 2\), we have \(\lim _{r \nearrow 1} \, K_r^{(\delta )} \, = \, +\infty \) for any \(\delta {\gt} 0\) small.

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

We have

We have

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

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

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

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

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

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

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

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

The simple random walk \(X= \big( X(t) \big)_{t \in \mathbb {N}}\) on the \(d\)-dimensional grid \(\mathbb {Z}^d\) is recurrent if \(d \le 2\) and transient if \(d \, {\gt} \, 2\).

The simple random walk \(X= \big( X(t) \big)_{t \in \mathbb {N}}\) on the \(d\)-dimensional grid \(\mathbb {Z}^d\) is expectation recurrent if \(d \le 2\) and expectation transient if \(d \, {\gt} \, 2\).