Brownian motion on spaces of discrete regular curves

TL;DR

This paper studies Brownian motion on spaces of discrete regular curves equipped with reparametrization-invariant Sobolev metrics $g^m$ and shows that, for $m\ge 2$, geodesic completeness implies stochastic completeness via Grigor'yan's volume-growth criterion. A discrete finite-dimensional model $\mathbb{R}_*^{d\times n}$ with metric $g^m$ is analyzed, establishing a volume bound $V_{g^m}(v_0,r) \le (2r)^{dn} D_0 e^{D_1 r}$ which, together with geodesic completeness, yields existence of Brownian motion for all times. The authors provide Euler–Maruyama simulations of Brownian paths to illustrate diffusion in these spaces and compare behavior across metric orders, highlighting a rigorous foundation for data statistics on shape spaces. Additionally, a heuristic analysis of the toy space of triangles modulo Euclidean motions suggests that stochastic completeness may extend to $m\in\{0,1\}$ in certain reduced models, offering avenues for future rigorous study.

Abstract

We introduce and study Brownian motion on spaces of discrete regular curves in Euclidean space equipped with discrete Sobolev-type metrics. It has been established that these spaces of discrete regular curves are geodesically complete if and only if the Sobolev-type metric is of order two or higher. By relying on a general result by Grigor'yan and controlling the volume growth of geodesic balls, we show that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, that is, the associated Brownian motion exists for all times. This provides a rigorous footing for performing data statistics, such as data inference and data imputation, on these spaces. Our result is the first stochastic completeness result in shape analysis that applies to the full shape space of interest. For illustrative purposes, we include simulations for sample paths of Brownian motion on spaces of discrete regular curves. For the space of triangles in the plane modulo rotation, translation and scaling, we further provide heuristics which suggest that this space remains stochastically complete even for Sobolev-type metrics of order zero and one.

Figures (3)

• Figure 1: Three examples of Brownian motion on the space of discrete regular curves with respect to the metric $g^2$. Top: Evolution of the centroid of each curve from $t=0$ to $t=10$ with step size $\Delta t =0.01$. Bottom: Corresponding trajectories of curves modulo translation. We plot every 10th curve corresponding to the centroid highlighted in red on the top plot. (a) Initialization at a square with $n=4$ points. (b) Initialization at a smooth curve with $n=100$. (c) Initialization at a shape from the MPEG7 dataset (lizard) with $n=100$.
• Figure 2: (a) Brownian motion on the space of discrete regular curves in $\mathbb{R}^2$ for $n=100$ with respect to 1st-, 2nd-, and 4th-order Sobolev-type metrics, initialized at the unit circle discretized with $100$ points. (b) Evolution of the minimum edge length over time for $10$ independent simulations from $t=0$ to $t=1000$ each with respect to $g^1$, $g^2$ and $g^4$.
• Figure 3: Plots of the conformal factors $f_0$, $f_1$ and $f_2$ corresponding to the restrictions of the metrics $g^0$, $g^1$ and $g^2$ to the space of triangles with two fixed vertices. We display the conformal factors on $[-2,2]\times[-2,2]$ as well as a small region around the singularity at $v_0=(1,0)$.

Theorems & Definitions (12)

• Theorem 1: Cerqueira--Hartman--Klassen--Bauer cerqueira2024sobolevmetricsspacesdiscrete
• Theorem 2
• Theorem 3: Grigor'yan grigoryan1grigoryan2
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Proposition 7