Visibility-Constrained Diffusion Process

• Visibility-constrained diffusion processes are stochastic models that integrate standard diffusion with explicit visibility restrictions, affecting sensitivity and control.
• The mathematical formulation augments classical reflected diffusions with visibility masks, projective mappings, and an augmented Skorohod framework to restrict active state spaces.
• These processes are applied in robotics, social networks, finance, and biology, where numerical and analytical techniques enforce geometric and measurement limitations.

A visibility-constrained diffusion process is a stochastic system whose state evolution is governed by diffusion (typically via SDEs or Markov chains), but where the dynamics and/or the observability of the state are restricted by explicit visibility constraints. These constraints may reflect geometric, measurement, or perception limitations, such that only certain components or regions of the state space are accessible, active, or subject to regulation. Visibility constraints manifest both in classical diffusion and in modern generative, planning, or learning contexts and fundamentally impact sensitivity, inference, and control properties of the process.

1. Mathematical Formulation and Classical Foundations

Visibility-constrained diffusion processes are formalized by augmenting standard constrained diffusion (such as reflected diffusions in a positive orthant) with visibility masks, partial observability, or projective mappings. In the classical setting studied by (Dieker et al., 2011), given a multidimensional reflected diffusion $Z_{\epsilon}$ constrained to the nonnegative orthant and drift parameter $\epsilon$, the pathwise sensitivity to infinitesimal changes in the drift is characterized by

$$a(t) = \lim_{\epsilon \to 0^+} \frac{Z(t) - Z_\epsilon(t)}{\epsilon}$$

which, in one dimension, yields an explicit form $a(t) = t - B(t)$, with $B(t) = \sup\{ s \in [0, t] : Z(s) = 0 \}$ (the last hitting time of the boundary).

In higher dimensions, the process $(z, y, a, b)$ solves an augmented Skorohod problem:

• $z(t) = x(t) + R y(t)$, $y(0) = 0$, $y$ nondecreasing, $\int_0^\infty z(t) dy(t) = 0$
• $a(t) = \chi(t) - \tilde{R}\, b(t)$ with $\chi(t) = t E$
• $b(0) = 0$, $b$ nondecreasing, $\int_0^\infty z(t) db^j(t) = 0$ (boundary complementarity)
• Additional: $a_i(t) = 0$ whenever $z_i(t) = 0$

Visibility constraints modify this system by restricting the active state space to a "visible" set, with the derivative process $a(t)$ forced to vanish not at the full boundary $\partial \mathbb{R}_+^J$ but instead at the visible boundary determined by application-dependent restrictions.

2. Visibility Constraints: Definitions and Extensions

Visibility can refer to multiple forms of access restriction:

• Geometric occlusion in robotics and planning: only spatial regions unobstructed by obstacles are visible, as formalized by the open segment criterion $$V(t) = \{ q \in \mathbb{R}^2 : (pq) \setminus \{p, q\} \not\in O \}$$ for obstacles $O$ (Knuth et al., 2020).
• Observability in networked systems: not all state components are measured or available for control/sensitivity analysis.
• Perceptual constraints: signal decay or limited attention in social contagion yields dynamic visibility masks (Hodas et al., 2012).

In diffusion modeling, visibility constraints require analytical machinery that correctly couples the evolution of observable state coordinates with the underlying stochasticity and ensures that derivatives, complementarity, and regulatory feedback are defined only on the visible subset.

3. Augmented Skorohod Problem and Sensitivity Analysis

The augmented Skorohod problem provides a rigorous mechanism to encode both physical and visibility constraints in diffusion:

Component	Mathematical Definition	Visibility-Constraint Adaptation
State $z$	$z(t) = x(t) + R y(t)$, $\int_0^\infty z(t) dy(t) = 0$	$z$ defined only on visible coordinates/faces
Regulator $y$	$y(0) = 0$, $y$ nondecreasing	$y$ increments only on visible boundaries
Derivative $a$	$a(t) = \chi(t) - \tilde{R} b(t)$, $\chi(t) = t E$	$a_i(t) = 0$ when $z_i(t)$ on (visible) boundary
Aux $b$	$b(0) = 0$, $b$ nondecreasing, $\int_0^\infty z(t) db^j(t) = 0$	$b$ increments only for visible coordinates

In settings where full observability does not hold, these conditions are projected onto the visible set, and "zero-set" vanishing of $a_i$ applies to the observed boundary.

The basic adjoint relationship (BAR) for stationary distributions adapts accordingly:

$$\int_{visible\;\Omega} [T f](z, a) d\pi(z, a) + ... = 0$$

where supports of measures and test functions are restricted to the visible set.

4. Impact of Visibility Constraints: Performance and Dynamics

Visibility constraints alter both local and long-run behavior:

• Sensitivity: As visibility restrictions reduce the active dimensionality, small drift perturbations affect only the fluid and jump components along observable directions. Stationary performance averages, such as expectations $\mathbb{E}[\phi(Z(\infty))]$ and their derivatives, become

$$\left.\frac{d}{d\epsilon} \mathbb{E}[\phi(Z_\epsilon(\infty))] \right|_{\epsilon=0} = \mathbb{E}[A(\infty) \phi'(Z(\infty))]$$

with $A$ the derivative process active only on visible boundaries.

• Regulatory feedback: The frequency and intensity of derivative jumps (e.g., in $A$) are increased or decreased depending on how the drift pushes the process against the visibility boundary.
• System control: Only observable or visible coordinates and boundaries are involved in regulation, with optimization and gradient-based tuning forced to restrict to that subset.

5. Applications and Generalizations

Visibility-constrained diffusion processes are essential in diverse domains:

• Robotics and path planning: Learning from demonstration with partial observability reconstructs obstacle maps via CSPs and cell decomposition, inferring safety and feasible paths in environments where only partial regions are observable (Knuth et al., 2020).
• Social networks: Information propagation constrained by visibility and attention decays, as measured by time-response functions $P_t(\Delta t)$ and friend-dependent normalization of propagation probability (Hodas et al., 2012).
• Quantitative finance and network queues: Many regulated systems must operate under partial observability, with sensitivity analysis restricted to visible accounts or nodes.
• Biological systems and protein design: Sampling in constrained molecular or conformational spaces where only physically meaningful (i.e., visible) states are permitted (Fishman et al., 2023).

These processes also admit generalizations to visibility-constrained manifolds, density-constrained laws (via penalization), and projected constraint sets.

6. Analytical and Numerical Techniques for Visibility Constraints

Analytical solution methods rely on the extension of reflection, regulation, and sensitivity machinery to the restricted visible set. Numerical approaches include:

• Penalization approximation (Jabir, 2017): Weak constraint enforcement via Wasserstein penalty, where the penalization drift is directed according to deviation from a visibility-constrained law or set $K$.
• Projection or barrier methods: Explicit Euclidean or Riemannian projections onto visible sets, or the addition of log-barrier (interior point) functions that blow up near non-visible boundaries.
• Constraint Satisfaction Problem solvers: In planning, cell decomposition and survey-cell graphs encode geometric visibility constraints, enabling inference of safe paths and environment maps from partial demonstrations.

7. Implications for Optimization, Control, and Inference

In the presence of visibility constraints:

• Sensitivity analysis provides gradient information only for the observable part, restricting optimization and robust control efforts to that subset.
• When the drift is perturbed, only the visible region’s boundary or interior responds, potentially pushing the system against previously inert facets.
• Both the continuous (fluid) and jump (boundary/face) parts of the derivative process $A$ reflect interactions modulated by visibility, highlighting the need for careful design of controllers and estimators in partially observed stochastic systems.

Visibility-constrained diffusion models thus extend classical regulated diffusion with foundational mathematical rigor, ensuring analytical, numerical, and practical tools address the unique challenges posed by partial observability, geometric occlusion, and measurement restrictions.