1.1. Hyperbolic Brownian motion

The hyperbolic plane $\mathbb{H}^2$ is the unique complete, simply connected two-dimensional Riemannian manifold with constant curvature $-1$ . A point $o \in \mathbb{H}^2$ is distinguished as the origin, and the hyperbolic distance $d_\mathbb{H}$ equips $\mathbb{H}^2$ with a norm and a metric. For distinct points $x, y \neq o$ , there are unique hyperbolic geodesics between o, x and between o, y, and these geodesics subtend an angle at o which is zero only if x lies on the geodesic from o to y, or vice versa. Every point $x \in \mathbb{H}^2 \setminus \{ o \}$ can be represented in geodesic polar coordinates $ x = (r,\theta)$ , where $r = d_\mathbb{H} (o, x) \in (0,\infty)$ , and $\theta \in [0,2\pi)$ is an angle relative to a fixed reference direction.

Hyperbolic Brownian motion is a stochastic process $B\,:\!=\,(B_t)_{t \in {\mathbb R}_+}$ on $\mathbb{H}^2$ , started from $B_0 \,:\!=\, o$ , whose dynamics can be described in geodesic polar coordinates $B_t = (R_t, \theta_t) \in {\mathbb R}_+ \times [0,2\pi)$ , $t > 0$ . The ${\mathbb R}_+$ -valued radial process $R \,:\!=\, (R_t)_{t \in{\mathbb R}_+}$ starts from $R_0\,:\!=\,0$ and satisfies the autonomous stochastic differential equation (SDE), in which $W^R$ is a standard $\mathbb{R}$ -valued Brownian motion,

(1) \begin{equation} \mathrm{d}R_t = \frac{\mathrm{d}t}{2\tanh R_t} + \mathrm{d}W_t^R \quad \text{for all } t \in {\mathbb R}_+.\end{equation}

Since $B_0 = o$ , the angle $\theta_0$ is not canonically defined, but for $t \geq s > 0$ the increment $\theta_t - \theta_s$ is described via an $\mathbb{R}$ -valued post-s winding process $\Theta^{(s)} \,:\!=\, (\Theta^{(s)}_t)_{t \geq s}$ which starts from $\Theta^{(s)}_s \,:\!=\,0$ and satisfies the SDE

(2) \begin{equation} \mathrm{d}\Theta^{(s)}_t = \frac{\mathrm{d}W^{(s)}_t}{\sinh R_t} \quad \text{for all } t \geq s,\end{equation}

where $W^{(s)}$ is a standard Brownian motion, independent of $W^R$ . Then $\theta_t = ( \theta_s + \Theta^{(s)}_t )$ modulo $2\pi$ . Combined with the entrance law $\theta_s \sim \mathrm{Unif}\mkern2mu [0,2\pi)$ , for every $s >0$ , inevitable thanks to the rapid spinning out from the origin [Reference Itô and McKean21, Section 7.16], the SDEs (1) and (2) provide a unique-in-law description. In particular, the law of B is rotationally invariant. This also gives a skew product decomposition of Brownian motion on $\mathbb{H}^2$ , similarly to Euclidean Brownian motion in $\mathbb{R}^d$ , $d \geq 2$ . Note that from (1) it can be shown that $R_t >0$ for all $t >0$ , almost surely (a.s.; see Lemma 4) so that in (2), $\sinh R_t > 0$ for $t>0$ . (Started instead from $B_0 \neq o$ , there is no issue with $\theta_0$ or rapid spinning, and (2) can be used to define the post-0 winding process $\Theta^{(0)}$ for all time.) For background on Brownian motion on $\mathbb{H}^2$ , including the above facts, we refer to [Reference Rogers and Williams34, Section V.36] and [Reference Gruet15, Section 2], for example.

The following result, part of which is well known (see Remark 1), shows that hyperbolic Brownian motion B exhibits ballistic transience and a random limiting direction. It quantifies the intuition, apparent from (1), that since $\lim_{x \to \infty} \tanh x = 1$ , a comparison for the process R at large scales is a Brownian motion with a constant drift $\frac12$ .

Proposition 1. Almost surely,

(3) \begin{equation} \text{sup}_{t \in{\mathbb R}_+} \Bigl| R_t - W_t^R - \frac{t}{2} \Bigr| < \infty \quad \textit{and} \quad \lim_{t\to\infty}\frac{R_t}{t} = \lim_{t \to \infty} \frac{\mathbb{E}\,R_t}{t} = \frac{1}{2}. \end{equation}

Moreover, there exists a random variable $\theta_\infty$ , with $\theta_\infty \sim \mathrm{Unif}\mkern2mu [0,2\pi)$ , and, for every $s >0$ , a random variable $\Theta_\infty^{(s)} \in \mathbb{R}$ , such that

(4) \begin{equation} \lim_{t \to \infty}\theta_t = \theta_\infty\quad \textit{a.s.} \quad \textit{and, for every }s >0,\quad \lim_{t \to \infty}\Theta^{(s)}_t = \Theta^{(s)}_\infty \quad \textit{a.s.} \end{equation}

Moreover, for every $s>0$ ,

(5) \begin{equation} \limsup_{t \to \infty} t^{-1} \log \bigl| \Theta^{(s)}_\infty - \Theta^{(s)}_t \bigr| = \limsup_{t \to \infty} t^{-1} \log | \theta_\infty - \theta_t | = -\frac{1}{2} \quad \textit{a.s.} \end{equation}

1.2. The convex hull and its perimeter length

The trajectory $B[0,t] \,:\!=\, \{ B_s \colon s \in [0,t] \}$ of hyperbolic Brownian motion up to time $t \in {\mathbb R}_+$ is a random compact subset of $\mathbb{H}^2$ containing the origin o; the subject of our work is the closed convex hull $\mathcal{H}_t \,:\!=\, \mathrm{conv}\,B[0,t]$ for $t \in {\mathbb R}_+$ , the smallest closed convex subset of $\mathbb{H}^2$ containing B[0, t]. Recall that a set $A \subseteq \mathbb{H}^2$ is convex if and only if, for every pair of distinct points $x,y \in A$ , the (unique) hyperbolic geodesic between x and y is contained in A. Then $\mathrm{conv}\,S$ , for $S \subseteq \mathbb{H}^2$ , is the intersection of all closed convex sets $A \subseteq \mathbb{H}^2$ with $S \subseteq A$ . Since $\mathbb{H}^2$ is locally convex and complete, and B[0, t] is compact, $\mathcal{H}_t$ is also compact. See Figure 1 for a simulation.

Figure 1.

Simulated hyperbolic Brownian motion trajectory and its convex hull. The SDEs (1) and (2) were each approximately solved, in turn, over time interval [0,10] with an Euler scheme with $10^6$ steps. The top left pane shows the R process and the top right pane shows the $\Theta^{(s)}$ process over time [s,10] for $s = 10^{-3}$ . The discrete approximation moving swiftly to a large negative value reflects the rapid spinning out from the origin. The bottom left pane shows the resulting Brownian trajectory $B_t$ (shown in red) over time $t \in [0,10]$ in a section of the Beltrami–Klein disk $\mathbb{D}_{\mathrm{K}}$ ; in this model geodesics are straight lines, so the convex hull ${\mathcal{H}}_{10}$ (boundary in blue) can be computed more easily. The bottom right pane shows that same trajectory (red) and convex hull (blue) represented in the Poincaré disk $\mathbb{D}_{\mathrm{P}}$ . See Section 2.1 for a summary of the different models of $\mathbb{H}^2$ and how to map between them. Note that already by time 10 the process appears very close to the boundary, so running longer simulations would yield little more visual information.

Figure 1 Long description

Panel A: A line graph shows the R process over time. The x-axis represents time (t) ranging from 0 to 10, and the y-axis represents the R process values ranging from 0 to 8. The line graph shows an upward trend with fluctuations. Panel B: Another line graph shows the Theta process over time. The x-axis represents time (t) ranging from 0 to 10, and the y-axis represents the Theta process values ranging from -10 to 0. The line graph shows a rapid decrease initially and then stabilizes. Panel C: A scatter plot shows the Brownian trajectory in the Beltrami-Klein disk model. The trajectory is depicted in red, and the convex hull boundary is shown in blue. Panel D: Another scatter plot shows the same Brownian trajectory and convex hull in the Poincaré disk model, with the trajectory in red and the convex hull in blue.

A compact, convex set $K \subset \mathbb{H}^2$ can be arbitrarily well approximated by convex (hyperbolic) polygons (i.e., convex hulls of finitely many points), and its boundary $\partial K$ has a well-defined perimeter length $\operatorname{perim} K$ . Consider the perimeter length of the hyperbolic Brownian convex hull: $L_t \,:\!=\, \operatorname{perim} \mathcal{H}_t$ for $t \in {\mathbb R}_+$ ; note that $L_0 = 0$ . In this paper we are interested in $\mathbb{E}\,L_t$ , and, particularly, in the asymptotics of $\mathbb{E}\,L_t$ as $t \to \infty$ . For sets $A_1 \subseteq A_2$ , the monotonicity property $\operatorname{perim}\operatorname{conv} A_1 \leq \operatorname{perim}\operatorname{conv} A_2$ holds (this can be checked using, e.g., the Cauchy formula of [Reference Alexander, Berg and Foote2] given at (17) in Section 2.2). The convex hull $\mathcal{H}_t$ certainly contains the geodesic line $\operatorname{conv} \{ o, B_t \}$ between the origin and the location of the process at time t, and hence its perimeter is at least twice the hyperbolic distance from the origin (this also follows from the Cauchy formula, as we explain in Example 1). In other words, $L_t \geq 2 R_t$ for all $t \in {\mathbb R}_+$ . An immediate consequence of Proposition 1 is thus that the perimeter length satisfies the ‘line segment’ lower bounds

(6) \begin{equation} \liminf_{t \to \infty}\frac{L_t}{t} \geq 1\quad \text{a.s.}, \qquad \liminf_{t \to \infty}\frac{{\mathbb{E}}\,L_{t}}{t} \geq 1.\end{equation}

Our main result gives an expression for ${\mathbb{E}}\,L_{t}$ in terms of an exponential functional of Brownian motion on the line, as well as large-time and small-time asymptotics for ${\mathbb{E}}\,L_{t}$ . Define

(7) \begin{equation} {\mathcal{E}}_t \,:\!=\, {\int}_0^t\exp\!(2W_s - s)\,\mathrm{d}s\quad \text{for } t\in {\mathbb R}_+,\end{equation}

where $W = (W_t)_{t \in {\mathbb R}_+}$ is a standard Brownian motion on $\mathbb{R}$ .

Theorem 1. $\mathbb{E}\,L_t = \sqrt{8\pi}\,\mathbb{E}\,\sqrt{\mathcal{E}_t}$ for every $t \in {\mathbb R}_+$ . Moreover,

(8) \begin{equation} \lim_{t \to 0}\frac{\mathbb{E}\,L_t}{\sqrt{8\pi t}\,} = 1, \qquad \lim_{t \to \infty}\frac{\mathbb{E}\,L_t}{2t} = 1. \end{equation}

Theorem 1 is a direct consequence of Theorem 2 combined with Proposition 2 below.

There has been a lot of work in the last couple of decades on exponential functionals such as $\mathcal{E}_t$ , motivated in part by actuarial or financial applications [Reference De Schepper and Goovaerts11, Reference Dufresne12]; see [Reference Matsumoto and Yor28] for a survey, e.g. [Reference Alili and Gruet3, Reference Yor41] for their links to hyperbolic Brownian motion, and e.g. [Reference Hariya and Yor17, Reference Yor41, Reference Yor42] for further properties. The asymptotics available for moments of $\mathcal{E}_t$ [Reference Hariya and Yor17] are the source of (8). On the other hand, the expectation identity we establish in Theorem 1 combined with results of [Reference Yor41] yield, for $\mathbb{E}\,L_t$ at each fixed $t \in {\mathbb R}_+$ , an exact, but complicated, multiple-integral expression (43), and it is not straightforward to even obtain a numerical estimate for $\mathbb{E}\,L_1$ (see Appendix A). However, a remarkable identity due to Yor [Reference Yor42] leads to an exact expression at a random exponential time; let ${\operatorname{Exp}\!(\lambda)}$ denote the exponential distribution with parameter $\lambda >0$ (hence mean $1/\lambda$ ).

Corollary 1. For $\lambda >0$ , let $T_\lambda \sim {\operatorname{Exp}\!(\lambda )}$ be independent of B. Then

(9) \begin{equation} \mathbb{E}\,L_{T_\lambda} = G(\sqrt{8\lambda+1}), \quad \text{where } G(x) \,:\!=\, \pi\bigg(\frac{x-1}{x+1}\bigg)\bigg(\frac{\Gamma(({x-1})/{4})}{\Gamma(({x+1})/{4})}\bigg)^2, \end{equation}

where $\Gamma$ denotes the (Euler) gamma function. In particular, we have $\lim_{\lambda \to \infty} \sqrt{\lambda} \mathbb{E}\,L_{T_\lambda} = \pi \sqrt{2}$ , $\lim_{\lambda \to 0} \lambda \mathbb{E}\,L_{T_\lambda} = 2$ , and, when $\lambda =1$ , $\mathbb{E}\,L_{T_1} = G(3) = \pi^2/2 \approx 4.93\ldots$

In the case of Euclidean Brownian motion on $\mathbb{R}^2$ , its perimeter length $L^{\mathrm{E}}_t$ enjoys an exact formula due to Letac and Takács [Reference Letac and Takács24],

(10) \begin{equation} \mathbb{E}\,L^{\mathrm{E}}_t = \sqrt{ 8 \pi t} \quad \text{for all } t \in {\mathbb R}_+,\end{equation}

from which it readily follows that if $T_\lambda \sim {\operatorname{Exp}\!(\lambda )}$ is independent of the Brownian motion, then $\mathbb{E}\,L^{\mathrm{E}}_{T_\lambda} = \sqrt{8\pi} \, \mathbb{E}\,\sqrt{T_\lambda} = \pi \sqrt{ 2 / \lambda}$ for $\lambda > 0$ ; in particular, when $\lambda =1$ , $\mathbb{E}\, L^{\mathrm{E}}_{T_1} = \pi \sqrt{2} \approx 4.44\ldots$ The appearance of $\sqrt{8\pi}$ in both Theorem 1 and in (10) is not merely a coincidence, but a reflection of the local-Euclidean heuristic at small times. This is in line with the fact that the heat kernel $p_t(x,y)$ of $\mathbb{H}^2$ has the small-time behaviour (which holds in general, see, e.g., [Reference Hsu19, Chapter 5]), recalling that $d_\mathbb{H}$ denotes the hyperbolic distance, $\lim_{t\to0} -2t\log p_t(x,y) = d_\mathbb{H}(x,y)^2$ , which resembles its Euclidean counterpart, especially when x and y are close; also taking $t \approx 0$ in the SDE description yields the same intuition, since, as $r\to 0$ , $\tanh r \sim r$ and $\sinh r \sim r$ , so (1) and (2) approximate the polar description of Brownian motion on $\mathbb{R}^2$ for $t \approx 0$ . See Section 1.3 for further comparison between the hyperbolic and Euclidean settings.

Theorem 1 is proved in Sections 3 and 4. To prepare for the proofs, in Section 2 we recall some appropriate hyperbolic geometry, introduce notation associated with models of $\mathbb{H}^2$ that we will refer to later, and present a Cauchy formula from [Reference Alexander, Berg and Foote2] that is a key tool. Then Section 3 turns to hyperbolic Brownian motion on the Poincaré half-plane model, including its SDE description (Section 3.1), its asymptotics (Section 3.2), and the relation to exponential functionals (Section 3.3). As mentioned above, the proof of Proposition 1 is given in Section 5; this section is essentially self-contained, using only the SDE (1), and can be read independently of Sections 2–4. Finally, Appendix A provides an exact formula for $\mathbb{E}\,L_t$ as mentioned below Theorem 1, and gives the proof of Corollary 1.

To conclude this introduction, in Section 1.3 we discuss some consequences of our main results and some open questions that they raise, and draw some contrasts between the behaviour of the hyperbolic Brownian convex hull and the case of Brownian motion in $\mathbb{R}^d$ , $d \in \mathbb{N}$ (hereafter referred to as the ‘Euclidean case’). Convex hulls of Euclidean random walks, Brownian motions, and Lévy processes (etc.) have been an active topic of study going back to Lévy [Reference Lévy25]; see [Reference Majumdar, Comtet and Randon-Furling27] for a survey, and also [Reference Akopyan and Vysotsky1, Reference Bang, González Cázares and Mijatović4, Reference Cranston, Hsu and March9, Reference Cygan, Sandrić, Šebek and Wade10, Reference Eldan13, Reference Molchanov and Wespi30, Reference Snyder and Steele36–Reference Wade and Xu40] for a selection of classical and more recent work. It is natural to investigate related phenomena for hyperbolic Brownian motion in view of recent interest in ‘hyperbolic stochastic geometry’, e.g. [Reference Betken, Hug and Thäle5, Reference Bode, Fountoulakis and Müller6, Reference Godland, Kabluchko and Thäle14, Reference Herold, Hug and Thäle18, Reference Otto and Thäle31, Reference Owada and Yogeshwaran32]. Other aspects of the large-scale geometry of hyperbolic Brownian motion have been studied in [Reference Gruet15, Reference Gruet16].

1.3. Discussion and open questions

One important contrast between the behaviour of hyperbolic and Euclidean Brownian motions is the limiting-direction result (4). Clearly, $\mathcal{H}_s \subseteq \mathcal{H}_t$ for $0 \leq s \leq t$ ; one consequence of (4) is that the monotone limit set $\mathcal{H}_\infty \,:\!=\, \cup_{t \in {\mathbb R}_+} \mathcal{H}_t$ does not occupy the whole space, i.e. $\mathbb{P} ( \mathcal{H}_\infty = \mathbb{H}^2 ) =0$ . This is in contrast to the (standard, driftless) Euclidean case, where the corresponding limit set is the whole of $\mathbb{R}^d$ ; note that this is a reflection of how the process explores the sphere of directions, rather than of compact-set recurrence/transience.

Euclidean Brownian motion with non-zero drift also has the limiting direction property, and for the same reason its limit convex hull is not the full space. Indeed, the Euclidean analogue of the strong-law behaviour in (3) means that the convex hull of Euclidean Brownian motion with drift resembles a line segment for large t, in a sense that is strong enough to deduce, for example, strong laws for the perimeter length; see [Reference Lo, McRedmond and Wallace26, Reference McRedmond and Wade29, Reference Snyder and Steele36] for laws of large numbers for the perimeter of the convex hull of Euclidean random walks with drift, and [Reference McRedmond and Wade29, Reference Wade and Xu39] for fluctuations.

Despite the fact that (by Proposition 1) hyperbolic Brownian motion also satisfies a strong law and has a limiting direction, one contribution of our main result is to show that this line-segment comparison breaks down in the hyperbolic case. Our Theorem 1 shows that for t large we have $\mathbb{E}\,L_t \approx 4\mathbb{E}\,R_t$ . This is twice as much as would be obtained if the convex hull of hyperbolic Brownian motion were indeed getting close to a hyperbolic line segment, corresponding to the fact that the constant in Theorem 1 is twice the lower bound provided by (6). Natural questions remain; the next would settle whether or not a law of large numbers holds for $L_t$ .

Partial progress on this question might be to first consider the following.

To address these questions, finer study of the fluctuations of the angle process $\theta_t$ , as partially quantified by (5), seems necessary (cf. Remark 2).

To give some further insight into the comparison with the Euclidean setting, we consider the scaled trajectory $u \mapsto (t^{-1}R_{ut},\theta_{ut})$ for $u \in [0,1]$ . By the triangle inequality we have

$$d_\mathbb{H}((t^{-1}R_{ut},\theta_{ut}),(u,\theta_\infty)) \leq |t^{-1}R_{ut} - u| + \sinh\!(u)|\theta_{ut} - \theta_\infty|.$$

From Proposition 1 it follows that $\lim_{t \to \infty} \text{sup}_{0\leq u\leq 1} | t^{-1}R_{ut} - u | = 0$ a.s. Furthermore, for $t >0$ ,

\begin{align*} & \text{sup}_{0\leq u\leq 1} \sinh\!(u)|\theta_{ut} - \theta_\infty| \\ & \qquad = \max\big\{\text{sup}_{0\leq u\leq t^{-1/2} }\sinh\!(u)|\theta_{ut} - \theta_\infty|, \text{sup}_{t^{-1/2}\leq u\leq 1} \sinh\!(u)|\theta_{ut} - \theta_\infty|\big\} \\ & \qquad \leq \max\big\{2\pi\sinh\!\big(t^{-1/2} \big),\sinh\!(1)\text{sup}_{t^{-1/2} \leq u\leq 1} |\theta_{ut} - \theta_\infty|\big\}.\end{align*}

Proposition 1 shows that $\theta_t \to \theta_\infty$ a.s., implying that the second term inside the maximum goes to 0. Since the first term also goes to 0 as $t \to \infty$ , we can collect everything to find that

$$\lim_{t\to\infty} \text{sup}_{0 \leq u \leq 1} d_\mathbb{H}\big(\big(t^{-1}R_{ut},\theta_{ut}\big),(u,\theta_\infty)\big) = 0.$$

This implies that as sets, $\{(t^{-1}R_{ut},\theta_{ut}) \colon 0 \leq u \leq 1\} \to \{(u,\theta_\infty) \colon 0 \leq u \leq 1\}$ in the Hausdorff distance. Following reasoning based on continuous mapping (cf. [Reference Lo, McRedmond and Wallace26]), this implies that

$$\lim_{t \to \infty}\operatorname{perim}\operatorname{conv}\{(t^{-1}R_{ut},\theta_{ut}) \colon 0 \leq u \leq 1\} = 2\quad \text{a.s.}$$

In the Euclidean setting, from here we can derive the behaviour of the perimeter of the convex hull of $\{(R_{ut},\theta_{ut}) \colon 0 \leq u \leq 1\}$ , since it is simply t times larger. However, since distance in hyperbolic space grows differently, in the hyperbolic setting no such implication is valid.

Finally, we raise the following question.