2. Notation and main results

Throughout, we adopt the convention that $0 \in \mathbb{N}$ , and denote by $X=(X_n)_{n \in \mathbb{N}}$ a discrete-time, time-homogeneous Markov chain whose state space is a Riemannian manifold M, with the Borel sigma algebra. We review the geometric concepts that we need; the reader unfamiliar with this material might consult, for example, [Reference Lee20]. Suppose that M has dimension d. If $x \in M$ is a point, then the tangent space at x, denoted $T_x M$ , is a d-dimensional real vector space whose elements may be viewed as ‘vectors tangent to M at x’. The tangent bundle TM is a manifold of dimension 2d whose points consist of pairs (x, v), where $v \in T_x M$ . The space $T_x M$ inherits an inner product space structure, denoted $\langle \,\cdot\, \rangle_x$ , from the Riemannian metric on M. Let $\text{Planes}(M,x)$ denote the collection of two-dimensional subspaces of $T_x M$ . Then the sectional curvature of M at x with respect to the plane $\pi \in \text{Planes}(M,x)$ , which we denote by $\text{sec}(x,\pi)$ , is a real number that may be calculated using the metric on M. If M is a sphere of radius r (in any dimension), then $\text{sec}(x,\pi)=r^{-2}$ for all choices of x and $\pi$ . We are interested in hyperbolic manifolds, where the sectional curvature is everywhere negative. Finally, the Riemannian exponential map $\exp_x :\, T_x M \rightarrow M$ (when it exists) sends the vector $v \in T_x M$ to the point in M given by starting at x and travelling along the geodesic determined by v a Riemannian distance of $\sqrt{\langle v,v \rangle_x}$ . We denote by $\textrm{Dist}_M(x,y)$ the Riemannian distance between $x \in M$ and $y \in M$ .

Under Assumption 1, the Riemannian exponential map $\exp_x :\, T_x M \rightarrow M$ exists and is a diffeomorphism for every $x \in M$ [Reference Jost14, Lemma 2.1.4]. For technical reasons, we make an arbitrary (but fixed) choice of origin $O \in M$ , and work with local notions of recurrence and transience. See the discussion following Definition 1 for how these local notions relate to (global) recurrence and transience, which do not depend on a choice of origin. Having chosen O, the function

\begin{equation*} e_{\text{rad}}(x) = \frac{\exp^{-1}_{x}\!(O)}{\sqrt{\langle \exp^{-1}_{x}\!(O),\exp^{-1}_{x}\!(O)} \rangle_{x}} \qquad (x \neq O) \end{equation*}

is well defined. We also define some sequences of random variables, indexed by $n \in \mathbb{N}$ :

\begin{align*} V^{(n)} & = \exp^{-1}_{X_n}\!(X_{n+1}), \\[2pt] D^{(n)}_{\text{tot}} & = \sqrt{\langle V^{(n)}, V^{(n)} \rangle_{X_n}}, \\[2pt] D^{(n)}_{\text{rad}} & = \left\{\begin{array}{l@{\quad}l} -\langle V^{(n)},e_{\text{rad}}(X_n) \rangle_{X_n} & \text{ if } X_n \neq O , \\[5pt] D^{(n)}_{\text{tot}} & \text{ if } X_n = O , \end{array}\right. \\[2pt] \Phi^{(n)} & = \left\{\begin{array}{l@{\quad}l} {D^{(n)}_{\text{rad}}}/{D^{(n)}_{\text{tot}}} & \text{ if } D^{(n)}_{\text{tot}} \gt 0 , \\[5pt] 0 & \text{ if } D^{(n)}_{\text{tot}} = 0. \end{array}\right.\end{align*}

The TM-valued process $(X_n,V^{(n)})$ is known as a geodesic random walk (GRW). The excursion theory of a particular GRW on hyperbolic space, and the large deviation theory of GRWs on general Riemannian manifolds, have been respectively studied in [Reference Cammarota and Orsingher3, Reference Kraaij, Redig and Versendaal18]. Geometrically, $D^{(n)}_{\text{tot}}$ is the total Riemannian distance between $X_n$ and $X_{n+1}$ , $e_{\text{rad}}(x)$ is the unit vector that ‘points from x to the origin’, $D^{(n)}_{\text{rad}}$ is the length of the radial component of $V^{(n)}$ , and $\Phi^{(n)}$ is the cosine of the angle between $V^{(n)}$ and $({-}e_{\text{rad}}(X_n))$ in $T_{X_n}M$ . We also define the radial process $(R_n)_{n \in \mathbb{N}}$ by $R_n = \textrm{Dist}_M(X_n,O)$ .

We introduce another piece of notation. If H is a (suitable) function from a subset of $\mathbb{R}^3$ to $\mathbb{R}$ , then we write $\mathbb{E}_x[H(D_{\text{tot}}, \Delta R, \Phi)]$ as shorthand for $\mathbb{E}\big[H(D_{\text{tot}}^{(n)}, \Delta_n R, \Phi^{(n)}) \mid X_n=x\big]$ , where $\Delta_n R \,:\!=\, R_{n+1}-R_{n}$ . This notation is unambiguous because X is Markov, and so the expression does not depend on n. We make the following assumptions on the chain X.

Assumption 2 also appears in [Reference Georgiou, Menshikov, Mijatović and Wade8]. Without it, we can construct trivial examples of recurrent chains by having a probability of $10^{-6}$ (say) of jumping to the origin, regardless of the current location of the chain. Assumption 3 is global in nature, but Proposition 1 gives local conditions on X that are sufficent for Assumption 3 to hold. Proposition 1 is proved in Section 4.

The notions of O-recurrence and O-transience are local, and also appear in [Reference Georgiou, Menshikov, Mijatović and Wade8], although there they are simply called ‘recurrent’ and ‘transient’. For us, ‘recurrent’ means that, for any open $U \subset M$ , X almost surely visits U infinitely often, and ‘transient’ means ‘not recurrent’. If X is O-transient for some choice of O (equivalently for all choices of O) then X is transient. If X is O-recurrent for some choice of O, and the chain is ‘irreducible’ (in some suitable sense), then we would expect X to be recurrent. For example, suppose that X is known to visit a neighbourhood $N_0$ of some origin O infinitely often almost surely, and has the property that, for every open neighbourhood N of M, there exist $m \in \mathbb{N}$ and $\delta >0$ such that $\inf_{x \in N_0} \mathbb{P}_x[\tau_N \leq m] \geq \delta$ , where $\tau_{N}\,:\!=\,\min\{n \in \mathbb{N}\,:\, X_n \in N \}$ . Then a Borel–Cantelli-type argument similar to [Reference Menshikov, Popov and Wade22, Example 2.3.20] reveals that X is recurrent. A full analysis of the relationship between O-recurrence and recurrence is technically involved since M is neither discrete nor countable; this is beyond the scope of this paper.

We introduce the remaining notation that we need to state our main result. Given $x \in M$ and a real number $d_{\text{tot}} \geq 0$ (we should think of $d_\textrm{tot}$ as an observation of $D^{(n)}_\textrm{tot}$ for some n), the real-valued functions $k_{\text{min}}$ and $k_{\text{max}}$ are defined to measure the extremes of the sectional curvature within a distance $d_{\text{tot}}$ of x. More precisely, we define

\begin{align*} k_{\text{min}}(x,d_\textrm{tot}) & = \inf_{\substack{y \in M\,:\, \textrm{Dist}_{M}(x,y) \leq d_\textrm{tot}, \\ \pi \in \text{Planes}(M,y)}} \sqrt{-\textrm{sec}(y,\pi)} , \\ k_{\text{max}}(x,d_\textrm{tot}) & = \sup_{\substack{y \in M\,:\, \textrm{Dist}_{M}(x,y) \leq d_\textrm{tot}, \\ \pi \in \text{Planes}(M,y)}} \sqrt{-\textrm{sec}(y,\pi)}.\end{align*}

By Assumption 1, $0<\kappa \leq k_\textrm{min}(x,d_\textrm{tot}) \leq k_\textrm{max}(x,d_\textrm{tot})$ for all $x, d_\textrm{tot}$ . Given $k>0$ , $d_\textrm{tot}>0$ , and $\phi \in [{-}1,1]$ , let G be the real-valued function

(1) \begin{equation} G(k,d_\textrm{tot},\phi) = \frac{1}{k}\log\!{(\!\cosh\!(k d_\textrm{tot}) + \phi \sinh\!(k d_\textrm{tot}))}.\end{equation}

We demonstrate in the proof of Proposition 3 that G is an asymptotic estimate, valid when R is much larger than $D_{\text{tot}}$ , for the increment $\Delta R$ on a manifold of constant curvature $-k^2$ . Finally, the notation $\textbf{1}_E$ , for an event E, refers to the indicator function that is equal to 1 on E and zero on the complement of E.

Theorem 1. Let M and X satisfy Assumptions 1, 2, and 3. Let $S(r) = \{ x \in M \,:\, \textrm{Dist}_M(O,x)=r \}$ , and let

\begin{align*} \underline{\nu}_1(r) &= \inf_{x \in S(r)} \mathbb{E}_x[G(k_\textrm{min}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)], \\ \overline{\nu}_1(r) &= \sup_{x \in S(r)} \mathbb{E}_x[G(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)], \\ \underline{\nu}_2(r) &= \inf_{x \in S(r)} \big\{ \mathbb{E}_x[G^2(k_\textrm{min}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)\textbf{1}_{\Delta R \geq 0}] \\ & \qquad \qquad + \mathbb{E}_x[G^2(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)\textrm{1}_{\Delta R < 0}] \big\}, \\ \overline{\nu}_2(r) &= \sup_{x \in S(r)} \big\{ \mathbb{E}_x[G^2(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)\textbf{1}_{\Delta R \geq 0}] \\ & \qquad \qquad + \mathbb{E}_x[G^2(k_\textrm{min}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)\textbf{1}_{\Delta R < 0}] \big\}. \end{align*}

1. (i) If $\liminf_{r \rightarrow \infty}\!(2r \underline{\nu}_1(r)-\overline{\nu}_2(r))>0$ then X is O-transient.

2. (ii) If instead $\liminf_{r \rightarrow \infty} \underline{\nu}_2(r)>0$ and there exist $r_0 \geq 0$ , $\theta>0$ such that

   \begin{equation*} 2r\overline{\nu}_1(r) \leq \bigg(1+\frac{1-\theta}{\log r}\bigg)\underline{\nu}_2(r) \qquad {for\ all\ } r \geq r_0 , \end{equation*}
   then X is O-recurrent.

To say that a chain X is zero drift is to say that, for all $x \in M$ , the conditional law of $X_{n+1}$ , given that $X_n=x$ , has x as its Riemannian centre of mass (or ‘barycentre’ [Reference Émery and Mokobodzki7]). This concept appears in the statistics and Monte Carlo literature as a way of taking means and medians on manifolds [Reference Arnaudon, Barbaresco and Yang2]. Zero-drift chains are also closely related to the notion of martingales on M. When $M=\mathbb{R}^d$ the two notions are equivalent, although for general M the story is slightly more complicated, as explained in [Reference Sturm28].

Recall from [Reference Georgiou, Menshikov, Mijatović and Wade8, Equation (1.11)] that a chain $(X_n)_{n \in \mathbb{N}}$ is called uniformly elliptic if there exists $\varepsilon>0$ such that

(3) \begin{equation} \mathbb{P}\big[\langle \exp^{-1}_{X_n}\!(X_{n+1}), w \rangle_{X_n} \geq \varepsilon \mid \mathcal{F}_n\big] \geq \varepsilon\end{equation}

almost surely for all unit vectors $w \in T_{X_n}M$ and all $n \in \mathbb{N}$ . In Euclidean space, zero-drift recurrent uniformly elliptic chains exist. By contrast, in Section 5 we derive the following consequence of Theorem 1.

We deduce Theorem 2 from a stronger result, namely that, under our assumptions, if a zero-drift chain is to be recurrent, then the function

(4) \begin{equation} Q(x)\,:\!=\,\mathbb{E}_x\big[D_\textrm{tot}^2-D_\textrm{rad}^2\big]\end{equation}

cannot remain bounded above zero as $\textrm{Dist}_M(x,O) \rightarrow \infty$ . Geometrically, Q is the total variance of the increment, conditional on the chain currently being at $x \in M$ , in the transverse direction. Some intuition as to why this stronger result is true is given at the start of Section 5. In the Euclidean case, Q provides a great deal of information as to the recurrence of a zero-drift chain: if we ignore certain boundary or degenerate cases, and assume that Q and $\mathbb{E}_x\big[D_\textrm{tot}^2\big]$ tend to limiting values when far from the origin, then these limiting values alone are enough to deduce recurrence or transience [Reference Georgiou, Menshikov, Mijatović and Wade8, Theorem 2.3]. By contrast, in the hyperbolic setting, rapid decay of Q is insufficient to imply recurrence, unless very strong assumptions are placed on the tails of the chain.

We now proceed as follows. Sections 3, 4, and 5 respectively prove Theorem 1, Proposition 1, and Theorem 2. Finally, Section 6 gives some examples.

3. Geometric calculations and proof of Theorem 1

For real numbers $k>0$ , $d_\textrm{tot} \geq 0$ , $\phi \in [{-}1,1]$ , and $x \in M$ , define a function F by

(5) \begin{equation} F(k,d_\textrm{tot},\phi,x) = \frac{1}{k} \textrm{arccosh} (\!\cosh{k r} \cosh{k d_\textrm{tot}} + \phi \sinh{k r}\sinh{k d_\textrm{tot}} ) - r ,\end{equation}

where $r=r(x)=\textrm{Dist}_M(x,O)$ . Proposition 2 implies that if the manifold has constant curvature $-k^2$ , then $F(k,D^{(n)}_\textrm{tot},\Phi^{(n)},x)$ is the exact value of the increment $R_{n+1}-R_{n}$ , given that $X_n=x$ . We stress that Proposition 2 is a purely geometric result; its proof does not require any probabilistic information.

Proposition 2. Let M be a manifold satisfying Assumption 1. For brevity, we write $k^{(n)}_\textrm{min}$ (or $k^{(n)}_\textrm{max}$ ) in place of $k_\textrm{min}(X_n,D^{(n)}_\textrm{tot})$ (or $k_\textrm{max}(X_n,D^{(n)}_{tot})$ ). Recall also that $\Delta_n R \,:\!=\, R_{n+1}-R_{n}$ . Then, for all $n \in \mathbb{N}$ ,

\begin{align*} F(k^{(n)}_\textrm{min},D^{(n)}_\textrm{tot},\Phi^{(n)},R_n) & \leq \Delta_n R \leq F(k^{(n)}_\textrm{max},D^{(n)}_\textrm{tot},\Phi^{(n)},R_n) ,\\[5pt]F^2(k_\textrm{min},D^{(n)}_\textrm{tot},\Phi^{(n)},R_n) \textbf{1}_{\Delta_n R \geq 0} + & F^2(k_\textrm{max},D^{(n)}_\textrm{tot},\Phi^{(n)},R_n) \textbf{1}_{\Delta_n R < 0} \\[5pt] & \leq (\Delta_n R)^2 \leq \\[5pt] F^2(k_\textrm{max},D^{(n)}_\textrm{tot},\Phi^{(n)},R_n) \textbf{1}_{\Delta_n R \geq 0} & + F^2(k_\textrm{min},D^{(n)}_\textrm{tot},\Phi^{(n)},R_n) \textbf{1}_{\Delta_n R < 0}. \end{align*}

Some routine algebraic manipulation of the first inequality in Proposition 2 results in the following corollary.

Corollary 1. A sufficient condition for $R_{n+1} - R_{n} < 0$ is that

\begin{equation*} \Phi^{(n)} < \frac{\coth{k_\textrm{max}^{(n)}R_n}\big(1-\cosh{k_\textrm{max}^{(n)}D^{(n)}_\textrm{tot}}\big)}{\sinh{k_\textrm{max}^{(n)}D^{(n)}_\textrm{tot}}}. \end{equation*}

In particular, for all $\varepsilon>0$ there exists a constant $r_\varepsilon$ such that, if $R_n>r_\varepsilon$ and

\begin{equation*} \Phi^{(n)} < \frac{(1-\varepsilon)\big(1-\cosh{k_\textrm{max}^{(n)}D^{(n)}_\textrm{tot}}\big)}{\sinh{k_\textrm{max}^{(n)}D^{(n)}_\textrm{tot}}} , \end{equation*}

then $R_{n+1}-R_{n}<0$ . A sufficient condition for $R_{n+1}-R_{n} \geq 0$ is that

\begin{equation*} \Phi^{(n)} \geq \frac{\coth{k_\textrm{min}^{(n)}R_n}\big(1-\cosh{k_\textrm{min}^{(n)}D^{(n)}_\textrm{tot}}\big)}{\sinh{k_\textrm{min}^{(n)}D^{(n)}_\textrm{tot}}}. \end{equation*}

In particular, for all $\varepsilon>0$ there exists a constant $r_\varepsilon$ such that, if $R_n>r_\varepsilon$ and

\begin{equation*} \Phi^{(n)} \geq \frac{(1+\varepsilon)\big(1-\cosh{k^{(n)}_\textrm{min} D^{(n)}_\textrm{tot}}\big)}{\sinh{k^{(n)}_\textrm{min}D^{(n)}_\textrm{tot}}} , \end{equation*}

then $R_{n+1}-R_{n} \geq 0$ .

Proposition 3. Assume that M has constant curvature $-k^2$ , and that X satisfies Assumption 2. Let $x \in M$ be a point of distance $R_x$ from O. Then

(6) \begin{equation} \mathbb{E}_x[\Delta R] = \frac{1}{k}\mathbb{E}_x[\!\log\!(\!\cosh k D_{\textrm{tot}}+\Phi \sinh k D_{\textrm{tot}})]+O(R_x^{1-p}) , \end{equation}

(7) \begin{equation} \mathbb{E}_x\big[(\Delta R)^2 \textbf{1}_{\Delta R \geq 0} \big] = \frac{1}{k^2}\mathbb{E}_x\big[\!\log^2(\!\cosh k D_{\textrm{tot}}+\Phi \sinh k D_{\textrm{tot}})\textbf{1}_{\Delta R \geq 0}\big]+O(R_x^{2-p}), \end{equation}

where the implicit constants in the remainder terms depend only on k, p, and B, and remain bounded as a function of k as $k \rightarrow \infty$ . Moreover, (7) remains true if $\textbf{1}_{\Delta R \geq 0}$ is changed to $\textbf{1}_{\Delta R<0}$ throughout.

Proof. For brevity, let $\alpha=\cosh{kR_x}$ , $\beta=\sinh{kR_x}$ , $c=\cosh{k D_{\text{tot}}}$ , and $s=\sinh{k D_{\text{tot}}}$ . Using Proposition 2, followed by some algebraic manipulation, we find that

(8) \begin{align} & \bigg| \Delta R-\frac{1}{k}\log\!{(c+\phi s)} \bigg| =\bigg| \frac{1}{k} \textrm{arccosh}{(\alpha c + \Phi \beta s)}-R_x-\frac{1}{k}\log\!{(c+\Phi s)} \bigg|\\& \quad = \bigg|\frac{1}{k} \log\!(2(\alpha c + \Phi \beta s)) + \frac{1}{k} \log\! \bigg(\frac{1+\sqrt{1-(\alpha c + \Phi \beta s)^{-2}}}{2}\bigg) - R_x - \frac{1}{k} \log\!(c+\Phi s)\bigg| \nonumber \\ & \quad = \bigg|\frac{1}{k} \log\! \bigg( \frac{2(\alpha c + \Phi \beta s) \textrm{e}^{-kR_x}}{c+\Phi s} \bigg) + \frac{1}{k} \log\! \bigg(\frac{1+\sqrt{1-(\alpha c + \Phi \beta s)^{-2}}}{2}\bigg)\bigg| \nonumber \\ & \quad = \bigg|\frac{1}{k} \log\! \bigg( \frac{c(1+\textrm{e}^{-2kR_x}) + \Phi s (1-\textrm{e}^{-2kR_x}) }{c+\Phi s} \bigg) + \frac{1}{k} \log\! \bigg(\frac{1+\sqrt{1-(\alpha c + \Phi \beta s)^{-2}}}{2}\bigg)\bigg| \nonumber \\ & \quad = \bigg| \frac{1}{k} \log\! \bigg(1+\frac{c-\Phi s}{c+\Phi s} \textrm{e}^{-2kR_x}\bigg)+\frac{1}{k}\log\! \bigg(\frac{1+\sqrt{1-(\alpha c + \Phi \beta s)^{-2}}}{2}\bigg)\bigg| \nonumber \\ & \quad \leq \bigg| \frac{1}{k} \log\! \bigg(1+\frac{c-\Phi s}{c+\Phi s} \textrm{e}^{-2kR_x}\bigg) \bigg| +\bigg| \frac{1}{k}\log\! \bigg(\frac{1+\sqrt{1-(\alpha c + \Phi \beta s)^{-2}}}{2}\bigg) \bigg|. \nonumber \end{align}

Note that, since $\alpha, \beta, c, s \geq 0$ and $\Phi \in [{-}1,1]$ ,

(9) \begin{equation} \textrm{e}^{-kD_{\text{tot}}} = c-s \leq c+s=\textrm{e}^{kD_{\text{tot}}}. \end{equation}

In particular, $c \pm \Phi s \geq 0$ . Further,

(10) \begin{equation} \alpha c + \Phi \beta s \geq \alpha c - \beta s = \cosh\!(k R_x - k D_{\text{tot}}) \geq \max\!\big(\tfrac{1}{2} \exp\!(kR_x - kD_{\text{tot}}), 1 \big). \end{equation}

We can verify that if $u \geq 0$ then $0 \leq \log\!(1+u) \leq u$ , and that if $u \geq 1$ then

\begin{equation*} \bigg\lvert \log\! \bigg( \frac{1+\sqrt{1-u^{-2}}}{2} \bigg) \bigg\rvert \leq \frac{1}{u^2}. \end{equation*}

It follows that

(11) \begin{align} \bigg| \Delta R-\frac{1}{k}\log\!{(c+\Phi s)} \bigg| &\leq \frac{1}{k} \frac{c-\Phi s}{c+\Phi s} \textrm{e}^{-2kR_x} + \frac{1}{k} \frac{1}{(\alpha c + \Phi \beta s)^2} \nonumber \\ &\leq \frac{1}{k} \textrm{e}^{2k(D_{\text{tot}}-R_x)} + \frac{4}{k} \textrm{e}^{2k(D_{\text{tot}}-R_x)} \quad \text{[by (9) and (10)]} \nonumber\\ &= \frac{5}{k} \textrm{e}^{2k(D_{\text{tot}}-R_x)}. \end{align}

Let E be the event that $D_{\text{tot}} \leq R/2$ , and $E^\textrm{c}$ its complement. Then

\begin{align*} \bigg|\mathbb{E}_x\bigg[ \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg] \bigg| & \leq \bigg|\mathbb{E}_x\bigg[ \bigg( \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg) \textbf{1}_E \bigg] \bigg| \\ & \quad + \bigg|\mathbb{E}_x\bigg[ \bigg( \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg) \textbf{1}_{E^\textrm{c}}\bigg] \bigg| \,=\!:\, Q_1 + Q_2. \end{align*}

Using (11),

\begin{align*} Q_1 \leq \mathbb{E}\bigg[ \frac{5}{k} \textrm{e}^{2k(D_{\text{tot}}-R_x)} \textbf{1}_E \bigg] \leq \mathbb{E}\bigg[\frac{5}{k}\textrm{e}^{-kR_x} \textbf{1}_E\bigg] \leq \frac{5}{k}\textrm{e}^{-kR_x}. \end{align*}

To bound $Q_2$ , we use (9) and then Assumption 2:

\begin{align*} Q_2 \leq \mathbb{E}_x[2 D_{\text{tot}} \textbf{1}_{E^\textrm{c}}] = \mathbb{E}_x\big[2 D_{\text{tot}}^p D_{\text{tot}}^{1-p} \textbf{1}_{E^\textrm{c}}\big] \leq \bigg(\frac{R_x}{2}\bigg)^{1-p} \mathbb{E}_x\big[2D_\textrm{tot}^p \textbf{1}_{E^\textrm{c}}\big] \leq 2^{p} R_x^{1-p} B. \end{align*}

Combining these bounds establishes (6). For (7), we note, from the elementary observation that $a^2-b^2 = (a-b)^2 + 2b(a-b)$ , that

\begin{align*} & \bigg|\mathbb{E}_x\bigg[ (\Delta R)^2 \textbf{1}_{\Delta R \geq 0} -\bigg(\frac{1}{k}\log\!{(c+\Phi s)}\bigg)^2 \textbf{1}_{\Delta R \geq 0} \bigg] \bigg| \leq \\ & \mathbb{E}_x\bigg[\bigg| \bigg( \bigg( \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg)^2 \textbf{1}_E + \frac{2}{k} \log\!(c+\Phi s) \bigg( \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg) \textbf{1}_E \bigg)\textbf{1}_{\Delta R \geq 0}\bigg| \bigg] \\ & + \mathbb{E}_x\bigg[\bigg| \bigg( \bigg( \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg)^2 \textbf{1}_{E^\textrm{c}} + \frac{2}{k} \log\!(c+\Phi s) \bigg( \Delta R -\frac{1}{k}\log\!{(c+\Phi s)} \bigg) \textbf{1}_{E^\textrm{c}} \bigg) \textbf{1}_{\Delta R \geq 0} \bigg| \bigg] \\ & \,=\!:\, Q_3 + Q_4. \end{align*}

Using (11), we find that

\begin{align*} Q_3 & \leq \mathbb{E}_x\bigg[\bigg(\bigg(\frac{5}{k}\textrm{e}^{2k(D_{\text{tot}}-R_x)}\bigg)^2+ 2D_{\text{tot}} \bigg(\frac{5}{k}\textrm{e}^{2k(D_{\text{tot}}-R_x)}\bigg)\bigg) \textbf{1}_E \bigg] \\[3pt] & \leq \mathbb{E}_x\bigg[ \frac{25}{k^2} \textrm{e}^{-2kR_x} + \frac{10}{k} \textrm{e}^{-kR_x} D_{\text{tot}}\bigg]. \end{align*}

Assumption 2 together with Lyapunov’s inequality implies that $\mathbb{E}_x[D_{\text{tot}}]$ is bounded as a function of x, giving a bound on $Q_3$ of the required form. To bound $Q_4$ , Assumption 2 gives

\begin{align*} Q_4 \leq \mathbb{E}_x\big[8 D_{\text{tot}}^2 \textbf{1}_{E^\textrm{c}} \big] = \mathbb{E}_x\big[8 D_{\text{tot}}^p D_{\text{tot}}^{2-p} \textbf{1}_{E^\textrm{c}}\big] \leq 2^{p+1}R^{2-p} \mathbb{E}_x\big[D_{\text{tot}}^{p} \textbf{1}_{E^\textrm{c}}\big] \leq 2^{p+1} B R^{2-p}, \end{align*}

from which we deduce (7). Finally, the same proof as above shows that (7) holds when $\textbf{1}_{\Delta R \geq 0}$ is changed to $\textbf{1}_{\Delta R<0}$ throughout.

3.1. Proof of Theorem 1

Recall that the functions F and G are defined in (5) and (1), respectively.

Proof. In what follows, C is a constant that depends only on $\kappa$ , p, and B, but may change from line to line. We note that

(12) \begin{align} \mathbb{E}_x[\Delta R] & \leq \mathbb{E}_x[F(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi,x)] \nonumber \\ & \leq \mathbb{E}_x[G(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)] + C r^{1-p} , \end{align}

where $r=\textrm{Dist}_M(x,O)$ . The first inequality here is from Proposition 2. The second follows from the proof of Proposition 3, noting that the right-hand side of (8) is exactly $|F(k,D_{\text{tot}},\Phi,x) - G(k,D_\textrm{tot},\Phi)|$ . Similarly, we see that

(13) \begin{align} \mathbb{E}_x[(\Delta R)^2] & \leq \mathbb{E}_x[F^2(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi,x)\textbf{1}_{\Delta R \geq 0}] \nonumber \\ & \quad + \mathbb{E}_x[F^2(k_\textrm{min}(x,D_\textrm{tot}),D_\textrm{tot},\Phi,R)\textbf{1}_{\Delta R < 0}] \nonumber \\ & \leq \mathbb{E}_x[G^2(k_\textrm{max}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)\textbf{1}_{\Delta R \geq 0}] \nonumber \\ & \quad + \mathbb{E}_x[G^2(k_\textrm{min}(x,D_\textrm{tot}),D_\textrm{tot},\Phi)\textbf{1}_{\Delta R < 0}] + C r^{2-p}. \end{align}

Define, for $i=1$ and $i=2$ ,

(14) \begin{align} \underline{\mu}_i(r) = \underline{\nu}_i(r) - C r^{i-p}, \qquad \overline{\mu}_i(r) = \overline{\nu}_i(r) + C r^{i-p}. \end{align}

Taking suprema of (12) and (13) over S(r), and infima of the corresponding lower bounds over S(r), it follows that

(15) \begin{equation} \underline{\mu}_i(R_n) \leq \mathbb{E}[(\Delta R_n)^i \: | \: \mathcal{F}_n] \leq \overline{\mu}_i(R_n) \end{equation}

almost surely for all n, where $\mathcal{F}_n$ is the sigma algebra generated by $R_1,\dots,R_n$ .

Suppose now that the assumptions in Theorem 1(i) hold. It follows from (14), together with the fact that $p>2$ , that $\limsup_{r \rightarrow \infty} \overline{\mu}_2(r)<\infty$ and $\liminf_{r \rightarrow \infty}\big(2r \underline{\mu}_1(r)-\overline{\mu}_2(r)\big)>0$ . Theorem 1(i) now follows from these two expressions and (15), together with [Reference Menshikov, Popov and Wade22, Theorem 3.5.1].

Suppose instead that the assumptions in Theorem 1(ii) hold for some constants $r_0 \geq 0$ and $\theta>0$ , where $\theta=2\theta^{\prime}$ . Then, with the constant C as given in (14),

\begin{align*} 2r \overline{\mu}_1 - \bigg(1+\frac{1-\theta^{\prime}}{\log r} \bigg)\underline{\mu}_2 & = 2r\overline{\nu}_1 - \bigg(1+\frac{1-\theta^{\prime}}{\log r} \bigg) \underline{\nu}_2+C\bigg(3+\frac{1-\theta^{\prime}}{\log r} \bigg)r^{2-p} \\ & = \bigg(2r\overline{\nu}_1 -\bigg(1+\frac{1-\theta}{\log r} \bigg) \underline{\nu}_2 \bigg) - \frac{\theta^{\prime} \underline{\nu}_2}{\log r} + C\bigg(3+\frac{1-\theta^{\prime}}{\log r} \bigg)r^{2-p}. \end{align*}

By assumption, $\liminf_{r \rightarrow \infty} \underline{\nu}_2>0$ , and therefore the second term of the last equality decays more slowly than the third as $r \rightarrow \infty$ . It follows that there exists $r^{\prime}_0$ such that, for all $r \geq r^{\prime}_0$ ,

\begin{equation*} 2r \overline{\mu}_1 - \left(1+\frac{1-\theta^{\prime}}{\log r} \right)\underline{\mu}_2 \leq 0. \end{equation*}

Theorem 1(ii) follows from this and (15), together with [Reference Menshikov, Popov and Wade22, Theorem 3.5.2].

4. Non-confinement

In this section we prove Proposition 1. Our strategy is to use martingale arguments to deduce Proposition 1 from a similar result in Euclidean space [Reference Georgiou, Menshikov, Mijatović and Wade8, Proposition 2.1]. Compared to that result, ours applies to a wider class of processes (not just martingales), but at the price of being a little more restrictive: we require $\mathbb{E}[D_{\text{rad}}^2]\geq \varepsilon$ as opposed to $\mathbb{E}[D_{\text{tot}}^2]\geq \varepsilon$ . Let us note also that other non-confinement criteria are known (one example is [Reference Menshikov, Popov and Wade22, Equation (3.10)]), and proving non-confinement is often straightforward in practice. Let $\mathcal{F}$ be the filtration that is naturally generated by X.

Proof. By Toponogov’s theorem, it is enough to prove this in the case where M is Euclidean. Using the cosine rule for triangles in $\mathbb{R}^2$ , we can show that

\begin{equation*} R_{n+1} = R_n \sqrt{1+\frac{2 D^{(n)}_{\text{rad}}}{R_n}+\frac{\big(D^{(n)}_{\text{tot}}\big)^2}{R_n^2}}. \end{equation*}

Since $D^n_{\text{tot}} \geq |D^n_{\text{rad}}|$ , we deduce that

(16) \begin{equation} R_{n+1} - R_{n} \geq R_n \bigg( \bigg[\bigg( 1 + \frac{D^{(n)}_\textrm{rad}}{R_n} \bigg)^2\bigg]^{1/2} - 1 \bigg) \geq D^{(n)}_\textrm{rad} , \end{equation}

and the result follows upon taking expectations.

In the remainder of this section, $Y=(Y_n)_{n \in \mathbb{N}}$ is a process in $\mathbb{R}^d$ adapted to $\mathcal{F}$ , such that $Y_0=0$ . The processes L and A are defined using the Doob decomposition of Y [Reference Williams30]. More precisely, the jth one-dimensional component of Y is given by $Y_n^j = L_n^j + A_n^j$ , where, for each j, $1 \leq j \leq d$ , $L^j$ is an $\mathcal{F}$ -martingale and $A^j$ is a predictable process, with the property that if $Y^j$ is a submartingale then $A^j$ is non-negative and increasing.

Proof. (i) This follows from the bounds on $\mathbb{E}[|\Delta L|^p]$ and $\mathbb{E}[|\Delta A|^p]$ , and the inequality $|x+y|^p \leq C_{d,p}(|x|^p + |y|^p)$ for vectors $x,y \in \mathbb{R}^d$ , where $C_{d,p}$ is a constant.

(ii) We check that $\mathbb{E}[\Delta_n Z \mid \mathcal{G}_n]= \mathbb{E}[\Delta_n L \mid \mathcal{G}_n] + (\Delta_n A) \mathbb{E}[ \xi_{n+1} \mid \mathcal{G}_n] = 0+0 = 0$ , and that, by part (i) and Lyapunov’s inequality, there is a constant B ^′′ such that $\mathbb{E}[|Z_n|] \leq \sum_{k=0}^{n-1} \mathbb{E}[|\Delta_k Z|] \leq n B^{\prime\prime} < \infty$ .

(iii) We calculate

\begin{align*} \mathbb{E}[|\Delta_n Z|^2 \mid \mathcal{G}_n]&= \sum_{j=1}^d \mathbb{E}\left[ (\Delta_n L^j + \xi_{n+1} \Delta_n A^j)^2 \mid \mathcal{G}_n\right] \\ &= \sum_{j=1}^d \mathbb{E}\left[ (\Delta_n L^j)^2 + 2 \xi_{n+1} \Delta_n A^j \Delta_n L^j + (\Delta_n A^j)^2 \mid \mathcal{G}_n \right] \\ &= \mathbb{E}[|\Delta_n L|^2 + |\Delta_n A|^2 \mid \mathcal{G}_n] , \end{align*}

and, similarly, $\mathbb{E}[(\Delta_n Y)^2 \mid \mathcal{G}_n]=\mathbb{E}[(\Delta_n L)^2 + (\Delta_n A)^2 \mid \mathcal{G}_n]$ .

Proposition 5. Suppose that Y is an $\mathbb{R}$ -valued submartingale. Assume that there exist $B \in \mathbb{R}_{+}$ and $p>2$ such that $\mathbb{E}[|\Delta_n L|^p \mid \mathcal{F}_n] \leq B$ and $\mathbb{E}[|\Delta_n A|^p \mid \mathcal{F}_n] \leq B$ for all n. Assume also that there exists $\varepsilon>0$ such that $\mathbb{E}[|\Delta_n Y|^2 \mid \mathcal{F}_n] \geq \varepsilon$ for all n, and that

(17) \begin{equation} \mathbb{P}\Big[\!\limsup_{n \rightarrow \infty} L_n > -\infty\Big] = 1 . \end{equation}

Then $\mathbb{P}[\!\limsup_{n \rightarrow \infty}{|Y_n| = \infty}]=1$ . In particular, if Y is bounded below almost surely then $\mathbb{P}[\!\limsup_{n \rightarrow \infty}{Y_n = \infty}]=1$ .

Proof. We work in the enlarged probability space described in Lemma 1. Let $Z=(Z_n)_{n \in \mathbb{N}}$ be as in Lemma 1. We claim that

(18) \begin{align} \Big\{ \omega \in \Omega \,:\, \limsup_{n \rightarrow \infty}{|Y_n| = \infty} \Big\} & \supseteq \Big\{ \omega \in \Omega \,:\, \limsup_{n \rightarrow \infty} {|Z_n|} = \infty \Big\} \nonumber \\ & \quad \cap \Big\{ \omega \in \Omega \,:\, \limsup_{n \rightarrow \infty} L_n > -\infty \Big\}. \end{align}

To see this, first suppose that $\omega \in \Omega$ is such that A is bounded, say $A_n \leq W$ for all n. Then $Y_n \geq Z_n - 2 W$ for all n. On the other hand, if A is unbounded, then, since A is positive and increasing, $\lim_{n \rightarrow \infty} A_n = \infty$ and so $\limsup_{n \rightarrow \infty} A_n + L_n$ will be infinity provided that there exists $W \in \mathbb{R}$ such that $L_n > W$ infinitely often. This establishes (18). By 17, it suffices to prove that $\limsup_{n \rightarrow \infty} |Z_n|=\infty$ almost surely. Lemma 1, combined with [Reference Georgiou, Menshikov, Mijatović and Wade8, Proposition 2.1], establishes this result.

5. Uniform ellipticity and proof of Theorem 2

In this section we prove Theorem 2, which states that any uniformly elliptic zero-drift chain must be transient. We first give an intuitive explanation for why this is true. Let M be a manifold of constant curvature $-k^2$ . Suppose that the chain is currently a distance $R \gg 1$ from the origin, and that it makes a purely transverse step of unit length (‘transverse’ means perpendicular to the geodesic joining the origin to the chain’s current location). Then the (Euclidean or hyperbolic) Pythagorean theorem reveals that the change in the chain’s distance from the origin is given by

\begin{equation*} \Delta R = \begin{cases} R \big( \sqrt{1+R^{-2}} - 1 \big) & \text{ if } k=0 , \\[4pt] k^{-1} \textrm{arccosh}{\left(\!\cosh kR \cosh k \right)} - R & \text{ if } k<0. \end{cases}\end{equation*}

Expanding this to first order in $R^{-1}$ , we see that as $R \rightarrow \infty$ , $\Delta R$ tends to a limit that is zero if $k=0$ , but positive if $k<0$ . For this reason, if the variance in the transverse direction remains bounded above zero, then we would expect a zero-drift chain to be transient. Theorem 2 makes this intuition precise.

Lemma 2. For all real numbers $k>0$ , $d_\textrm{tot} \geq 0$ , and $\phi \in [{-}1,1]$ ,

\begin{equation*}d_{\textrm{rad}} + J_\textrm{min}(k,d_{\textrm{tot}})(d_{\textrm{tot}}^2-d_{\textrm{rad}}^2) \leq G(k,d_\textrm{tot},\phi) \leq d_{\textrm{rad}} + J_\textrm{max}(k,d_{\textrm{tot}})(d_{\textrm{tot}}^2-d_{\textrm{rad}}^2),\end{equation*}

where

\begin{align*} d_\textrm{rad} &= \phi d_\textrm{tot}, \\ J_\textrm{min}(k,d_{\textrm{tot}}) &= \frac{1}{2d_{\textrm{tot}}^2}\bigg( d_{\textrm{tot}} - \frac{\sinh\!(k d_{\textrm{tot}})}{k(\!\cosh\!(k d_{\textrm{tot}})+\sinh\!(k d_{\textrm{tot}}))} \bigg), \\ J_\textrm{max}(k,d_{\textrm{tot}}) &= \frac{1}{2d_{\textrm{tot}}^2}\bigg( {-}d_{\textrm{tot}} + \frac{\sinh\!(k d_{\textrm{tot}})}{k(\!\cosh\!(k d_{\textrm{tot}})-\sinh\!(k d_{\textrm{tot}}))} \bigg). \end{align*}

Moreover, $J_\textrm{min}$ is positive, increasing in k, and decreasing in $d_{\textrm{tot}}$ , whereas $J_\textrm{max}$ is non-negative and increasing in both k and $d_{\textrm{tot}}$ .

Proof. For fixed k and $d_{\textrm{tot}}$ , consider the function

\begin{equation*} H(\phi) \,:\!=\, \bigg(\frac{1}{k} \log\!(\!\cosh\!(kd_{\text{tot}})+\phi \sinh\!(k d_{\text{tot}}))-\phi d_{\text{tot}} \bigg)(1-\phi^2)^{-1}. \end{equation*}

It is lengthy but elementary to check that H is decreasing on $\phi \in [{-}1,1]$ and that its limits at $\phi=\pm 1$ are

\begin{equation*} \frac{1}{2}\bigg( \pm d_{\text{tot}} \mp \frac{\sinh\!(k d_{\text{tot}})}{k(\!\cosh\!(kd_{\text{tot}}) \pm \sinh\!(kd_{\text{tot}}))}\bigg). \end{equation*}

The first part follows, and the remainder follows from a direct check.

In the following theorem, the function Q is as defined in (4).

Proof. Let $c=\cosh\!(\kappa D_{\text{tot}})$ , $s=\sinh\!(\kappa D_{\text{tot}})$ , and A be a constant to be chosen later. Using Lemma 2, we obtain

\begin{align*} \mathbb{E}_x[G(\kappa,D_\textrm{tot},\Phi)] & \geq \mathbb{E}_x[D_{\text{rad}}] + \frac{1}{2} \mathbb{E}_x\bigg[\frac{1}{D_{\text{tot}}^2}\bigg( D_{\text{tot}}-\frac{s}{\kappa(c+s)} \bigg)\big(D_{\text{tot}}^2-D_{\text{rad}}^2\big) \bigg] \\ & \geq \frac{1}{2} \mathbb{E}_x\bigg[\frac{1}{D_{\text{tot}}^2}\left( D_{\text{tot}}-\frac{s}{\kappa(c+s)} \right)\big(D_{\text{tot}}^2-D_{\text{rad}}^2\big)\textbf{1}_{D_{\text{tot}}<A} \bigg] \\ & = \frac{1}{2} \mathbb{E}_x\bigg[\bigg( \frac{1}{D_{\text{tot}}} - \frac{1}{2 \kappa D_{\text{tot}}^2}(1-\textrm{e}^{-2 \kappa D_{\text{tot}}}) \bigg)\big(D_{\text{tot}}^2-D_{\text{rad}}^2\big)\textbf{1}_{D_{\text{tot}}<A} \bigg] \\ & \geq \frac{1}{2} \mathbb{E}_x\bigg[\bigg( \frac{1}{A} - \frac{1}{2 \kappa A^2}(1-\textrm{e}^{-2\kappa A}) \bigg)\big(D_{\text{tot}}^2-D_{\text{rad}}^2\big)\textbf{1}_{D_{\text{tot}}<A} \bigg], \end{align*}

where the last line follows from the fact that the function

\begin{equation*}\frac{1}{d_{\text{tot}}}-\frac{1}{2 \kappa d_{\text{tot}}^2}(1-\textrm{e}^{-2 \kappa d_{\text{tot}}})\end{equation*}

is positive and decreasing in $d_{\text{tot}}$ . Therefore, there is a constant $A_0$ , depending only on $\kappa$ , such that, if $A>A_0$ and $\textrm{Dist}_M(O,x) \geq d_{\text{min}}$ , then

\begin{align*} \mathbb{E}_x[G(\kappa,D_\textrm{tot},\Phi)] & \geq \frac{1}{4A} \mathbb{E}_x\big[\big(D_{\text{tot}}^2-D_{\text{rad}}^2\big)\textbf{1}_{D_{\text{tot}}<A} \big] \\ & = \frac{1}{4A} \big( \mathbb{E}_x\big[D_{\text{tot}}^2-D_{\text{rad}}^2\big] - \mathbb{E}_x\big[\big(D_{\text{tot}}^2-D_{\text{rad}}^2\big) \textbf{1}_{D_{\text{tot}}>A}\big] \big) \\ & \geq \frac{1}{4A} \big( \varepsilon - \mathbb{E}_x\big[D_{\text{tot}}^2 \textbf{1}_{D_{\text{tot}}>A}\big] \big). \end{align*}

By Assumption 2,

\begin{align*} \mathbb{E}_x\big[D^2_{\text{tot}} \textbf{1}_{D_\textrm{tot}<A}\big] = \mathbb{E}_x\big[D^p_\textrm{tot} \cdot D^{2-p}_\textrm{tot} \textbf{1}_{D_\textrm{tot}>A}\big] \leq \mathbb{E}_x[D_\textrm{tot}^p] A^{2-p} \leq B A^{2-p}. \end{align*}

Choose A sufficiently large that $B A^{2-p} \leq {\varepsilon}/{2}$ . This then gives

(19) \begin{equation} \mathbb{E}_x[G(\kappa,D_\textrm{tot},\Phi)] \geq \frac{\varepsilon}{2}. \end{equation}

Equation (2) therefore holds, and X is O-transient.

Proof of Theorem 2.Conditional on $X_n = x$ , view $v(x) = \exp_{x}^{-1}\!(X_{n+1})$ as a random vector in $T_x M$ . Decompose v(x) as $v=\lambda_{\text{rad}} e_{\text{rad}}(x)+w(x)$ , where $\lambda_{\text{rad}} \in \mathbb{R}$ and w(x) is a random vector orthogonal to $e_{\text{rad}}(x)$ . For each $\omega \in \Omega$ , w(x) is uniquely defined and does not depend on a choice of basis for $T_x M$ , because O is fixed and uniquely determines $e_{\text{rad}}(x)$ . We may now interpret Q(x) geometrically as $Q(x)=\mathbb{E}_x[\langle w(x), w(x) \rangle_x]$ , the expected squared length of the transverse component of v.

Choose a unit-length vector $e_{\text{trans}}(x)$ perpendicular to $e_\textrm{rad}(x)$ . By the Cauchy–Schwarz inequality, $\langle w, w \rangle_x \geq \langle w, e_{\text{trans}}(x) \rangle_x^2 = \langle v, e_{\text{trans}}(x) \rangle_x^2$ , and hence

(20) \begin{equation} Q(x) \geq \mathbb{E}_x[\langle v, e_{\text{trans}}(x) \rangle^2]. \end{equation}

If X is uniformly elliptic in the sense of (3) then there exists $\varepsilon>0$ such that $\mathbb{P}_x[\langle v, e_{\text{trans}}(x) \rangle^2_x \geq \varepsilon^2] \geq \varepsilon$ for every $x \in M$ . It follows from (20) that $Q(x) \geq \varepsilon^3$ for all x, and X is O-transient by Theorem 3.

Remark 2. We finish this section with a comment on the rate of escape of transient processes. Assumption 2 and the triangle inequality imply that there is a constant C such that $\mathbb{E}_x[\Delta R] \leq C$ for all x. Therefore, under our Assumptions 1, 2, and 3, [Reference Menshikov and Wade23, Theorem 2.3] (with $\beta=0$ ) shows that there exists a constant $\Lambda$ such that, almost surely,

(21) \begin{equation} \limsup_{n \rightarrow \infty} \frac{R_n}{n} \leq \Lambda. \end{equation}

In the case of a zero-drift, uniformly elliptic chain, (19) together with the same theorem from [Reference Menshikov and Wade23] gives a constant $\lambda$ such that $\lambda \leq \liminf_{n \rightarrow \infty} \frac{X_n}{n}$ almost surely, which in combination with (21) gives a linear rate of escape. As mentioned in the introduction, it is a general theme in the literature that ‘hyperbolic’ processes tend to escape at linear speed. However, as we shall see in the next section, by reducing the transverse component when far from the origin (thereby losing uniform elipticity) we may reduce the rate of escape and eventually obtain recurrence.

6. Examples

Let V be a finite-dimensional inner product space of dimension d. Given $v \in V$ and $a,b>0$ , define $L_V(a,b,v)\,:\, V \rightarrow V$ to be the linear transformation that sends v to $av \sqrt{d} $ and any $w \in \langle v \rangle^\perp$ to $b w \sqrt{d}$ . Define an elliptical measure $\xi_V(a,b,v) \,:\, \text{Borel}(V) \rightarrow \mathbb{R}_{\geq 0}$ by $\xi_V = \mu_V \circ L_V^{-1}(a,b,v)$ , where $\mu_V$ is the uniform measure on the unit sphere in V. Thus, $\xi_V$ is supported on an ellipsoid with principal axes of lengths $a\sqrt{d},b\sqrt{d},\dots,b\sqrt{d}$ . Given a d-dimensional manifold M with origin $O \in M$ , and functions $a, b\,:\, M \rightarrow \mathbb{R}_{\geq 0}$ , define a measure $\mu_p\,:\, \text{Borel}(M) \rightarrow \mathbb{R}$ at each point $p \in M$ by $\mu_p = \xi_{T_p M}(a(p),b(p)$ , $e_{\text{rad}}(p) ) \circ \exp_p^{-1}$ , where $e_{\text{rad}}(O)$ is defined arbitrarily as some fixed unit-length vector in $T_O M$ (as far as recurrence and transience is concerned, this choice is unimportant). This defines what we refer to as the elliptic Markov chain with parameters a and b. In the case where a and b are constant, and M is Euclidean space, the elliptic Markov chain reduces to the example in [Reference Georgiou, Menshikov, Mijatović and Wade8, Section 3].

We claim that, for the elliptic Markov chain,

(22) \begin{align} \mathbb{E}_p[D_{\text{tot}}^2] = a(p)^2 + (d-1)b(p)^2, \qquad \mathbb{E}_p[D_{\text{rad}}^2] = a(p)^2.\end{align}

To prove this, note that the computation in [Reference Georgiou, Menshikov, Mijatović and Wade8, p.7] establishes this result when V has the Euclidean inner product. The general result follows from the definition of $\xi_{V}$ together with the fact that any two inner product spaces of dimension d whose inner products are positive definite are isometric.

We study a special case that allows us to give a recurrence criterion that relates the asymptotic behaviour of b and the curvature of the manifold. Choose $a(r)=a$ for all r, where $a>0$ is a positive constant. For constants $c>0$ and $\gamma \geq 0$ , choose the curvature of M such that, if p is a point at a distance r from O, then

\begin{equation*} \sec\!(p,\pi) = \begin{cases} -c^2 & \text{ if } r \leq 1 , \\-(c r^\gamma)^2 & \text{ if } r \geq 1 \end{cases}\end{equation*}

for all $\pi \in \textrm{Planes}(M,p)$ . Since $a>0$ , it follows from Proposition 1 that Assumption 3 holds. Further, assume that $b(r) \leq b$ for all r. Then Assumption 2 holds, because, almost surely,

(23) \begin{equation} D_\textrm{tot}^{(n)} \leq d_\textrm{max}\end{equation}

for all $n \in \mathbb{N}$ , where $d_\textrm{max}\,:\!=\, \sqrt{d} \max{(a,b)}$ . To apply Theorem 1, we need to compute $\underline{\nu}_1$ , $\underline{\nu}_2$ , $\overline{\nu}_1$ , and $\overline{\nu}_2$ . Although this could be done by choosing coordinates and writing down a multidimensional integral, we instead obtain estimates using only (22) and (23). This better enables comparison with the results in [Reference Georgiou, Menshikov, Mijatović and Wade8].

Since M is radially symmetric, we may unambiguously write $\mathbb{E}_r$ to mean $\mathbb{E}_x$ for any $x \in S(r)$ . Using the fact that $\mathbb{E}_r[D_\textrm{rad}]=0$ , together with (22) and (23), we obtain that, for all sufficiently large r,

\begin{align*} \underline{\nu}_1(r) &= \mathbb{E}_r[G(k_\textrm{min}(r,D_\textrm{tot}),D_\textrm{tot},\Phi)] \\ &\geq \mathbb{E}_r[J_\textrm{min}(k_\textrm{min}(r,D_\textrm{tot}),D_\textrm{tot})(D_\textrm{tot}^2-D_\textrm{rad}^2)] \\ &\geq J_\textrm{min}(k_\textrm{min}(r,d_\textrm{max}),d_\textrm{max}) \mathbb{E}_r[(D_\textrm{tot}^2-D_\textrm{rad}^2)] \\ &= J_\textrm{min}(k_\textrm{min}(r,d_\textrm{max}),d_\textrm{max}) (d-1) b(r)^2 \\ &= J_\textrm{min}(c (r-d_\textrm{max})^{\gamma},d_\textrm{max}) (d-1) b(r)^2 .\end{align*}

Similarly, $\overline{\nu}_1(r) \leq J_\textrm{max}(c(r+d_\textrm{max})^\gamma,d_\textrm{max}) (d-1) b(r)^2$ . To bound $\overline{\nu}_2$ and $\underline{\nu}_2$ , we simply observe that, for all $k>0$ , $\mathbb{E}_r[D_{\text{rad}}^2 \textbf{1}_{D_{\text{rad}}>0}] \leq \mathbb{E}_r[G(k,D_{\text{tot}},\Phi)^2] \leq \mathbb{E}_r[D_{\text{tot}}^2]$ and hence, by symmetry, $\frac{1}{2}a^2 \leq \underline{\nu}_2(r) \leq \overline{\nu}_2(r) \leq a^2 + (d-1)b(r)^2$ . We discuss the cases $\gamma=0$ and $\gamma>0$ separately.

Suppose that $\gamma=0$ and that, for sufficiently large r, $b(r)={b}/{r^\beta}$ for some constant b. Then the elliptic Markov chain is recurrent if $\beta>\frac{1}{2}$ and transient if $\beta < \frac{1}{2}$ . If $\beta=\frac{1}{2}$ , then the chain is transient provided $2J_\textrm{min}(c,d_\textrm{max})(d-1)b^2 > a^2$ , and recurrent provided $2J_\textrm{max}(c,d_\textrm{max})(d-1)b^2<\frac{1}{2} a^2$ . Now suppose that $\gamma>0$ . Note that, for fixed d, $J_\textrm{min}(d,k) \rightarrow {1}/{2d}$ as $k \rightarrow \infty$ , and $J_\textrm{max}(d,k) \leq \textrm{e}^{2kd}$ for all sufficiently large k, and so $J_\textrm{max}(d,(r+d)^\gamma) \leq \textrm{e}^{3dr^\gamma}$ for all sufficiently large r. Accordingly, we obtain transience if $({1}/{d_\textrm{max}}) r (d-1) b^2(r) > a^2$ and recurrence if $2r\textrm{e}^{3d_\textrm{max}r^\gamma} (d-1) b^2(r) < \frac{1}{2} a^2$ for all sufficiently large r.

Tighter estimates of $\underline{\nu}_i$ and $\overline{\nu}_i$ would give sharper criteria than those stated above, but we do not pursue this here as our main intention is to contrast the Euclidean and hyperbolic cases. Figure 1 shows numerical simulations of this example in the hyperbolic plane ( $c=1$ and $\gamma=0$ ). Only the third simulation, where the tranverse component decays to zero, shows recurrence. If analogues of these chains were constructed in Euclidean space, both the second and the third would be recurrent, since results in [Reference Georgiou, Menshikov, Mijatović and Wade8] show that, even if we take a and b to be constant, we can still obtain recurrence provided $2a>b$ .

Figure 1. Simulations of the elliptic Markov chain with parameters a(r) and b(r) in the hyperbolic plane. The law of each simulation in the upper row is given schematically by the corresponding picture in the lower row. We take $a(r)=a$ and $b(r)=\frac{b}{r^\beta}$ , where the constants $(a,b,\beta)$ respectively take the values $(0.01,0.2,0)$ , $(0.2,0.01,0)$ , and $(0.2,0.01,1.1)$ .

A result in [Reference Lenz, Sobieczky and Woess21] shows that if $k_\textrm{min}(r) \geq C r^{2+\varepsilon}$ for constants $C,\varepsilon>0$ then M is stochastically incomplete, and so the discussion above gives a recurrent chain on such a manifold, as promised. At an intuitive level, we might say that the chain resembles Brownian motion on the manifold locally (for example, it has zero drift), but has very different global properties.

Finally, we note that in the previous example, for a constant-curvature manifold (i.e. $\gamma=0$ ), we obtain recurrence provided $\sup_{x \in S(r)}Q(x)$ decays faster than $O({1}/{r})$ . We stress that this is not sufficient in general, as exemplified below.

Proof. We give an example of such a chain. Take the probability density of $D_{\text{tot}}$ , conditional on the chain being at $x \in M$ , to be the same for every x and given by $f_\textrm{tot}(y \mid x) = \frac{m-1}{y^m}$ , $1 \leq y < \infty$ , where m is a constant; it is necessary to choose $m>3$ in order for Assumption 2 to hold. For some function $\lambda(r)$ (to be chosen later), let

\begin{equation*} \varepsilon(y)= \frac{1-\cosh\!(y)+\sinh\!(y)}{\sinh\!(y)} \cdot \textbf{1}_{y \geq \lambda(r)}\end{equation*}

and, conditional on $D_{\text{tot}}=y$ , let $\Phi$ be distributed as

\begin{equation*} \Phi(\, \cdot \mid D_{\text{tot}}=y)= \begin{cases} 1 & \text{ with probability } \alpha(y) , \\[3pt] -1+\varepsilon(y) & \text{ with probability } 1-\alpha(y) , \\ \end{cases} \end{equation*}

where

\begin{equation*} \alpha(y) = \frac{1-\varepsilon(y)}{2-\varepsilon(y)} = \begin{cases} \frac{1-\cosh\!(y)+\sinh\!(y)}{2} & \text{ if } y \geq \lambda(r) , \\[3pt] \frac{1}{2} & \text{ otherwise}. \end{cases} \end{equation*}

Notice that $\varepsilon$ and $\alpha$ depend on the point $x \in M$ via its distance from O, although for brevity we omit this from our notation. We can check that $0 \leq \alpha(y) \leq 1$ for all $y \geq 1$ , so that this definition makes sense. The choice of $\alpha$ ensures that $\mathbb{E}[\Phi \mid D_{\text{tot}}=y]=0$ for all y, and hence that $\mathbb{E}_x[D_{\text{rad}}]=\mathbb{E}_x[\Phi D_\textrm{tot}]=0$ for all $x \in M$ . The choice of $\varepsilon$ is made to simplify some of the forthcoming expectation calculations. Having specified the distributions of $D_{\text{rad}}$ and $D_{\text{tot}}$ , it is straightforward to choose the transverse components to give a zero-drift chain. We compute

\begin{align*} \mathbb{E}_x[D^2_{\text{tot}}(1-\Phi^2) \mid D_{\text{tot}}=y] & = y^2 \mathbb{E}_x[1-\Phi^2 \mid D_{\text{tot}}=y ] \\ & = y^2(0+(1-\alpha)(1-({-}1+\varepsilon)^2)) = y^2 \varepsilon. \end{align*}

From now on, assume that $\lambda(r) \geq 1$ for all r. Then

(24) \begin{align} \mathbb{E}_x[D_{\text{tot}}^2(1-\Phi^2)] &= \int_{y=1}^\infty f(y) \mathbb{E}_x\big[D_{\text{tot}}^2(1-\Phi^2) \mid D_{\text{tot}}=y\big] \, \textrm{d} y \nonumber \\ & = \int_{y=\lambda}^\infty \frac{(m-1)}{\sinh\!(y)y^{m-2}} (1-\cosh\!(y)+\sinh\!(y)) \, \textrm{d} y \nonumber \\ & \leq \int_{y=\lambda}^\infty \frac{2m}{\textrm{e}^y y^{m-2}} \, \textrm{d} y \nonumber \\ & \leq \int_{y=\lambda}^\infty 2m \textrm{e}^{-y} \, \textrm{d} y = 2m \textrm{e}^{-\lambda}. \end{align}

On the other hand, we find that, if $y \geq \lambda(r)$ , $\mathbb{E}_x[G(1,D_\textrm{tot},\Phi) \mid D_{\text{tot}}=y] =\alpha \log\!(\!\cosh{y}+\sinh{y}) + (1-\alpha)\log\!(1) = y \alpha(y)$ , whereas if $y < \lambda(r)$ then $\mathbb{E}[G(1,D_\textrm{tot},\Phi) \mid D_{\text{tot}}=y] = 0$ . So,

(25) \begin{align} \mathbb{E}_x[G(1,D_\textrm{tot},\Phi)] &= \int_{y=\lambda}^\infty y \alpha(y) f(y) \, \textrm{d} y \nonumber \\ & = \int_{y=\lambda}^\infty \frac{(m-1)(1+\sinh\!(y)-\cosh\!(y))}{2y^{m-1}} \, \textrm{d} y \nonumber \\ & \geq \int_{y=\lambda}^\infty \frac{m-1}{4y^{m-1}} \, \textrm{d} y = \frac{m-1}{4(m-2)} \frac{1}{\lambda^{m-2}}. \end{align}

Choose $\lambda(r)=r^{({1}/({m-1}))}$ . Then (24) tells us that $\sup_{x \in S(r)} Q(x)$ has the required rate of decay, and (25) tells us that (2) holds and so the chain is transient.

7. Future work

In this section we briefly outline some further questions that might be considered. We might allow the curvature of the manifold M to be asymptotically zero (but always negative). Since recurrence and transience typically only depends on what happens very far from the origin, it is a priori not obvious whether chains will typically behave in a hyperbolic manner, in a Euclidean manner, or somewhere in between. There has been recent interest in random processes on manifolds whose metric (and so curvature) changes in time [Reference Coulibaly-Pasquier5, Reference Paeng24]. The latter article contains geometric conditions on the evolution of M for Brownian motion on M to be stochastically complete, and it would be interesting to see what effect (if any) these conditions have on the range of behaviours observed in discrete chains defined on M.

We have considered here only the radial process that measures distance from the origin, but for a full understanding we should also consider the angular process. As mentioned in the introduction, there is a general theme that transient processes on hyperbolic manifolds converge to a limiting angle. A concrete question is whether a zero-drift transient Markov chain in hyperbolic space, under suitable assumptions on the moments of its increments, must converge to a limiting angle. In cases where it does, it is natural to ask for a characterisation of the law of that angle.

In Euclidean space, the radial and angular processes of the scaling limits of a class of Markov chains similar to those found in this paper have been considered in [Reference Georgiou, Mijatović and Wade9]. The authors presented a stochastic differential equation satisfied by the limit, and described the behaviour of both the radial and angular components of the limit in detail. The notion of diffusive scaling generalises easily to the manifold setting [Reference Jørgensen13], and so we expect that in our case we would obtain manifold-valued diffusions with similar properties to those in [Reference Georgiou, Mijatović and Wade9]. In view of the qualitative differences between the Euclidean and hyperbolic cases, it would be of interest to compare the limits obtained for hyperbolic and Euclidean manifolds.