6 Treatment of the integral in the Fourier inversion
In this part, we analyze the integral
where \(\widehat{\mathrm{G}}_{r} \colon \mathbb {R}^d \to \mathbb {C}\) is the Fourier transform of the regularized Green’s function of a random walk \(X\) on \(\mathbb {Z}^d\). By Corollary 5.7, the finiteness of this integral in the limit \(r \nearrow 1\) characterizes expectation transience of \(X\).
6.1 Decomposition of the integral
The main integral \(I_r\) can be decomposed into two parts: an easy "high frequency part", which contains the contributions of \(\theta \) away from \(0\), and a more interesting "low frequency part", which contains the contributions of \(\theta \) near \(0\).
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 \(\vec{0} \in \mathbb {R}^d\).
For any \(0 {\lt} \delta \le \pi \) we can write
Obvious, since \([-\pi ,\pi ]^d= ([-\pi ,\pi ]^d\setminus B_\delta ) \cup B_\delta \) is a disjoint union.
6.2 Dominated convergence away from the origin
TODO: Define non-degenerate step distribution (essentially \(\sum _{u\in \mathbb {Z}^d} p(u) e^{\mathrm{i}u \cdot \theta } \ne 1\) for \(\theta \ne 0\) modulo periodicity).
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 \(\mathbb {R}\)).
Under the nondegeneracy conditions, on the compact set \([-\pi ,\pi ]^d\setminus B_\delta \), the continuous integrand \(\theta \mapsto \widehat{\mathrm{G}}_{r}(\theta )\) is bounded (and therefore dominated by a constant function) and it has the pointwise limit \(\lim _{r \nearrow 1} \widehat{\mathrm{G}}_{r}(\theta ) = \widehat{\mathrm{G}}_{1}(\theta )\). It therefore follows from the dominated convergence theorem that \(\lim _{r \nearrow 1} J_r^{(\delta )} = J_1^{(\delta )} \in \mathbb {R}\).
6.3 Monotone convergence near the origin
TODO: Think about the best conditions for step distribution under which monotone convergence can be applied (real-valuedness requires symmetricity of the step-distribution?!?).
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 \([0,+\infty ]\).
…
We can now rephrase the recurrence criterion in terms of only the low frequency integral.
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 \(\lim _{r \nearrow 1} \, K_r^{(\delta )} \, {\lt} \, +\infty \).
…
6.4 Characterizing finiteness of the integral for simple random walk
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
Positivity estimates and Taylor series approximation of the Fourier transform of the Green’s function given explicitly in Lemma 5.4. …
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.
By the increasing limit part of Lemma 6.6 and monotone convergence theorem, we have
By the second part of Lemma 6.6, \(\Re \mathfrak {e}\big( \widehat{\mathrm{G}}_{1}(\theta ) \big) \ge C_1 \| \theta \| ^{-2}\) (for some positive constant \(C_1\)), so using this lower bound in integration and switching to radial coordinates, we have
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.
By the increasing limit part of Lemma 6.6 and monotone convergence theorem, we have
By the second part of Lemma 6.6, \(\Re \mathfrak {e}\big( \widehat{\mathrm{G}}_{1}(\theta ) \big) \le C_2 \| \theta \| ^{-2}\) (for some positive constant \(C_2\)), so using this upper bound in integration and switching to radial coordinates, we have