Boundary Random Walks

• Boundary Random Walks are stochastic processes that highlight boundary interactions using trace processes, Dirichlet forms, and scaling limits.
• They utilize potential theory, martingale problems, and spectral analysis to bridge discrete random walks with continuum diffusions.
• Their applications range from hyperbolic groups and convex cones to reflected random walks and interactive image segmentation.

Boundary Random Walks are a broad class of stochastic processes in which the interaction with the boundary—whether a group-theoretic boundary at infinity, a geometric barrier in a domain, or a discrete set in a Markov state-space—plays a central analytic and probabilistic role. The study of boundary random walks encompasses trace processes at infinity on hyperbolic groups, discrete-to-continuum invariance principles on the half-line, reflected and absorbing behaviors in domains, as well as applications in applied domains such as interactive image segmentation. Approaches include Dirichlet-form analysis, martingale problem techniques, potential theory, and explicit spectral/functional-analytic constructions. This article systematically develops the foundational model classes, analytic formalisms, scaling limits, and representative applications of boundary random walks across discrete and continuous frameworks.

1. Dirichlet-Form and Potential-Theoretic Constructions on Group Boundaries

For Gromov-hyperbolic groups $G$ with a symmetric, finite second-moment generating probability measure $\mu$, the associated random walk induces rich boundary phenomena. The Hilbert-Dirichlet form on $\ell^2(G)$,

$$\mathcal{E}(f,f) = \frac12 \sum_{x,y\in G} \mu(x^{-1}y)(f(x)-f(y))^2$$

with domain $\ell^2(G)$ is regular and yields a Hunt process $X_t$ (Mathieu et al., 2018). The discrete Laplacian $Lf(x)=\sum_y \mu(x^{-1}y) f(y) - f(x)$ splits the domain via Royden decomposition into $\ell^2(G)$ and the space $\mathcal{N}$ of $\mu$-harmonic finite-energy functions.

Almost every path converges to a boundary point $Z_\infty\in\partial G$; the induced harmonic measure $\nu$ captures the law of $Z_\infty$. The boundary Dirichlet form (trace form) is obtained via time-change or Poisson extension,

$$\mathcal{E}_\partial(u,u) = \iint_{\partial G\times\partial G} (u(\xi)-u(\eta))^2\, \Theta(\xi,\eta)\, d\nu(\xi) d\nu(\eta),$$

where $\Theta$ is the Naïm kernel. Harmonic functions with finite energy on $G$ correspond to the boundary process via this trace, producing a Markov jump process on $\partial G$ with infinitesimal generator

$$(\mathscr{L}_\partial u)(\xi) = \int_{\partial G} (u(\eta)-u(\xi))\Theta(\xi,\eta)\,d\nu(\eta).$$

Analytically, the process can be reflected at the compactification $\bar G=G\cup\partial G$ through an extension of the domain, leading to a regular Dirichlet form on $L^2(\bar G,\omega)$ for full-support $\omega$, linking interior and boundary behavior via the Revuz correspondence and the associated positive continuous additive functionals.

This framework naturally produces a scale of Besov spaces on $\partial G$, parametrized either by the Naïm kernel or by Ahlfors-regular metrics. The key structural result is that these random-walk-induced Besov spaces are canonically Banach-isomorphic among themselves and to classical cohomological constructions in the conformal gauge.

Existence and regularity of the boundary process are secured under conformal dimension $<2$ for $\partial G$, ensuring the uniqueness and regularity of the Dirichlet forms. Harmonic measures and Ahlfors-regular measures are shown to belong to the class $S_0(\partial G)$ of finite energy-integral measures. Explicit integral criteria characterize the smoothness of harmonic measures in terms of the geometry of $\partial G$ (Mathieu et al., 2018).

2. Discrete Boundary Random Walks and Scaling Limits to Feller Diffusions

On the half-line or the discrete set $\mathbb{N}_\Delta$, boundary random walks (BRWs) are Markov chains with symmetric nearest-neighbor transitions in the interior and general boundary transition rules at $0$—including killing, reflection, stickiness, and boundary-initiated jumps (Li et al., 31 Dec 2025).

Under diffusive rescaling ($B_t^{(n)}=n^{-1} X^{(n)}_{\lfloor n^2 t\rfloor}$), suitably chosen families of boundary transition probabilities $\mathfrak{p}_j^{(n)}$ yield convergence in Skorokhod space $D([0,\infty))$ to one-dimensional Feller processes on $[0,\infty)$ defined by

$\mathcal{L}f(x) = \tfrac12 f''(x),\; x>0,$

with general Feller boundary condition

$$p_1 f(0) - p_2 f'(0) + \tfrac{p_3}{2} f''(0) + \int_{(0,\infty)} [f(0)-f(x)] p_4(dx) = 0$$

for prescribed parameters $(p_1,p_2,p_3,p_4)$. This encompasses absorbed (Dirichlet), reflected (Neumann), sticky (Wentzell), elastic (Robin), and jump-in boundary behaviors. The scaling limit is established rigorously via martingale problem methods, with quantitative error control in generator and local time approximations (Arroyave et al., 14 Jul 2025, Li et al., 31 Dec 2025).

The derived framework provides a versatile discrete approximation scheme for all Feller-classified Brownian motions with semipermeable boundaries, and explicit regime-matching shows how discrete parameters $(\alpha,\beta,A,B)$ or $(\mathfrak{p}_j^{(n)})$ map to the continuous boundary law.

3. Martin Boundary and Harmonic Function Theory for Killed Random Walks

For random walks killed upon exiting convex cones in $\mathbb{Z}^d$, boundary random walks are characterized via their Martin boundary. For zero-drift, full-rank, aperiodic walks, the killed Green function $G_K(x,y)$ exhibits distinct asymptotic regimes (Duraj et al., 2020):

• If $y$ remains distant from the boundary $\partial K$, $G_K(x,y) \sim V(x) u(y) / \|y\|^{2p+d-2}$, where $u$ is the (continuous) harmonic homogeneous function for the cone, $V$ its discrete analog.
• If $y / \|y\| \to \theta \in \partial K$, boundary behavior is controlled by passage into tangent cones at $\theta$.

In both regimes, all positive discrete harmonic functions are scalar multiples of $V(x)$, and the Martin boundary consists of a single point: conditioning to "stay in the cone forever" is unique. This minimality of the boundary random walk's Martin boundary is a sharp contrast to nonzero-drift or multidimensional cases, where the boundary can have rich structure (Duraj et al., 2020, Raschel et al., 2018, Lecouvey et al., 2015).

Applications include Doob $h$-transform constructions of conditioned walks and connections to Brownian motion in cones, where analogous uniqueness holds for harmonic functions and boundary limits.

4. Reflected and Absorbing Boundary Random Walks in Domains

Boundary behavior in planar and multidimensional domains includes cases with reflecting or absorbing boundary conditions and Robin (mixed) types. Reflecting random walks are modeled by modification of transition kernels at the boundary, leading to nontrivial recurrence and absorption regimes depending on drift, covariance, and local geometry (Menshikov et al., 2020, Kaldasch et al., 2021, Boccardo et al., 2018).

For random walks in the quarter-plane, the law of the number of visits to boundaries converges to explicit superpositions of geometric distributions, and functional equations for generating functions yield closed forms via the compensation approach. Coupling arguments and recursive distributional equations provide insights into the exact statistics of boundary contacts, even in singular or combinatorially constrained settings (Hoang et al., 7 Jul 2025).

Sharp spectral results and explicit formulas for the occupation-time distribution in finite intervals under Neumann or Robin boundary conditions quantify the evolution from bimodal (arcsine law) to unimodal "stiff" distributions as time increases (Kaldasch et al., 2021, Boccardo et al., 2018).

In random environments, boundary localization phenomena are governed by rate-function gaps between annealed and quenched large deviations at the boundary. In dimensions $d\leq3$, localization holds generically, but a true phase transition occurs for $d\geq4$ as random environment disorder is tuned (Bazaes, 2019).

5. Functional Analytic and Martingale Problem Implementations

Boundary random walks are unified at the analytic level via Dirichlet forms, positive sub/supermartingale constructions, and explicit generator extensions. For processes approximating continuum diffusions, tightness and identification of limit points are achieved by solving associated martingale problems with boundary terms, requiring precise control over local times at the boundary and compensation for discretization errors (Arroyave et al., 14 Jul 2025, Li et al., 31 Dec 2025).

For image analysis and machine learning, the Boundary Random Walks (BRW) algorithm strategically manipulates boundary nodes in graph-based random-walk schemes, sharpening uncertainty at object contours and yielding a self-reinforcing system for segmentation potential minimization. Iterative implementations (IBRW) leverage positive feedback to incrementally minimize ambiguous pixels, outperforming classical segmentation algorithms empirically (Xie et al., 2018).

6. Group-Theoretic Boundary Random Walks and Poisson Boundaries

In group and geometric settings, identification of boundary behavior is framed via the Poisson–Furstenberg boundary, which encodes maximal entropy end-behavior for random walks. Techniques such as Kaimanovich’s strip criterion, geometric compactification, and coupling of geometric projections play central roles in classifying the maximal (boundary) $\mu$-space, often aligning the boundary theory with tree boundaries (Bass–Serre theory) or ends of non-amenable groups (Cuno et al., 2015, Vershik et al., 2015).

The boundary random walk formalism extends naturally to quantum groups and tensor categories, where noncommutative analogs of the Martin boundary exist. Notions of transience, Green functions, and boundary convergence are formulated in the categorical setting, with stability under monoidal equivalence and compatibility with the commutative/classical theory (Jordans, 2016).

7. Selected Tabular Overview of Key Boundary Random Walk Settings

Model Setting	Boundary Mechanism	Limiting Object/Class
Symmetric RW on hyperbolic group $G$	Gromov boundary $\partial G$	Jump process on $\partial G$
Discrete-time RW on $\mathbb{N}_\Delta$	Killing, stickiness, jumps at $0$	Feller process on $[0,\infty)$
RW in quadrant or convex cone	Absorption at axes/facets	Unique harmonic function, trivial Martin boundary
Planar RW with reflecting or Robin BC	Drift, reflection, Robin reaction	Generalized boundary local time distributions
RW in random environment, conditioned at boundary	Local time, localization/delocalization	Phase transition in occupation statistics
Image segmentation via BRW/IBRW	Edge probability sharpening	Sharper segmentation with fewer seeds

• (Mathieu et al., 2018) Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms
• (Li et al., 31 Dec 2025) From boundary random walks to Feller's Brownian Motions
• (Arroyave et al., 14 Jul 2025) Scaling limit of boundary random walks: A martingale problem approach
• (Duraj et al., 2020) Martin boundary of random walks in convex cones
• (Raschel et al., 2018) Boundary behavior for random walks in cones
• (Hoang et al., 7 Jul 2025) Boundary contacts for reflected random walks in the quarter plane
• (Kaldasch et al., 2021) Stiffness of random walks with reflecting boundary conditions
• (Boccardo et al., 2018) A second order scheme for a Robin boundary condition in random walk algorithms
• (Cuno et al., 2015) Random walks on Baumslag-Solitar groups
• (Bazaes, 2019) Localization at the boundary for conditioned random walks in random environment in dimensions two and higher
• (Xie et al., 2018) An Iterative Boundary Random Walks Algorithm for Interactive Image Segmentation

This multidimensional corpus establishes both the generality and the technical depth of boundary random walks as a foundational tool across probability, analysis, combinatorics, and applications.