Transience of continuous-time conservative random walks

TL;DR

This work studies two continuous-time generalizations of conservative random walks in $\mathbb{R}^d$ driven by a time-inhomogeneous Poisson clock. Model A uses orthogonal axis directions, while Model B uses random flights on the sphere, with switching rates $\lambda(t)$ that decay as $t^{-\alpha}$ ($0<\alpha\le 1$) or as $(\ln t)^{-\beta}$ ($\beta>2$). The authors prove transience in dimensions $d\ge 2$ under these rate laws by analyzing the embedded walk $W_n$ and, for Model B, its 2D projection via spectral methods and Bessel function bounds; they employ PPP tail estimates, large-deviation techniques, and Borel-Cantelli arguments to show finitely many visits to any bounded region a.s. The results extend understanding of memory-rich, time-inhomogeneous continuous-time random walks and establish conditions under which such processes escape to infinity, with implications for related random-flight models and non-homogeneous Markov dynamics.

Abstract

We consider two continuous-time generalizations of conservative random walks introduced in [J.Englander and S.Volkov (2022)], an orthogonal and a spherically-symmetrical one; the latter model is known as {\em random flights}. For both models, we show the transience of the walks when $d\ge 2$ and the rate of changing of direction follows power law $t^{-α}$, $0<α\le 1$, or the law $(\ln t)^{-β}$ where $β>2$.

Figures (4)

• Figure 1: Model A: recurrence of conservative random walk on $\mathbb{R}^2$
• Figure 2: Model B: $\hat{W}_n$, (the projection of) conservative random walk on $\mathbb{R}^2$
• Figure 3: A ray from $\hat{W}_n$ passes through the unit circle
• Figure 4: $|J_0(x)|$ (red) and its upper bound $G(x)$ (blue).

Theorems & Definitions (32)

• Definition 1
• Remark 1
• Lemma 1: KE
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 3
• proof : Proof of Lemma \ref{['lemm1']}