Sticky-Threshold & Skew Diffusions

• Sticky-threshold and skew diffusions are stochastic processes characterized by singular interfaces that induce directional bias (skewness) and positive dwell time (stickiness).
• They are formulated via singular stochastic differential equations incorporating local times and measure-valued coefficients to capture reflection, absorption, and regime-switching phenomena.
• These models are pivotal for analyzing complex systems in heterogeneous media, optimal stopping problems, and physical interfaces, including networks and fractal boundaries.

Sticky-threshold and skew diffusions are a class of stochastic processes that generalize classical diffusions by introducing singular interfaces, typically modeled as thresholds or membranes, which may impart skewness (bias in directional crossing), stickiness (positive occupation time at the interface), or both. These mechanisms encode sharp local changes in dynamics and are motivated by applications ranging from interface phenomena in physical systems and heterogeneous media to singular controls and optimal stopping theory. The precise mathematical framework uses singular SDEs with local times, time-changed constructions, and piecewise or measure-valued coefficients, supporting a rich spectrum of interface behaviors, including reflection, absorption, skew passage, and sticky sojourn.

1. Structural Definitions and SDE Formulations

Sticky-threshold and skew diffusions are typically formulated as solutions to SDEs with singular coefficients, interpreted via symmetric local times. For a generic interface at a threshold $\theta$, a one-dimensional model is

$$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$

where $L^x_t(X)$ is symmetric local time at $x$, and $\nu$ is a measure encoding interface effects:

• Skew interface: $\nu(dx) = \beta\,\delta_\theta(dx)$, $\beta \in (-1,1)$, so that at $\theta$, $X$ switches direction with bias set by $\beta$;
• Sticky interface: Upon hitting $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$0, $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$1 spends a positive Lebesgue time at $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$2, implemented by requiring $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$3 for stickiness $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$4;
• Threshold diffusion: $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$5 and $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$6 may jump at $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$7.

In multidimensional or more complex geometries (e.g., prefractal interfaces (Capitanelli et al., 2014), star graphs (Berry et al., 2024)), these constructs generalize to vector-valued SDEs, with local time accumulated at specified sets or vertices, and interface rules encoded in transmission, reflection, absorption, or sticky terms.

The concise SDE for a sticky-oscillating-skew (SOS) threshold at $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$8 is: $$X_t = X_0 + \int_{\mathbb{R}} L^x_t(X)\,\nu(dx) + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dB_s,$$9 with stickiness $L^x_t(X)$0 and skewness $L^x_t(X)$1 (Anagnostakis et al., 2024).

2. Interface Regimes and Parameterizations

Interface behavior is controlled by the singular coefficients:

• Pure reflection: $L^x_t(X)$2 (total local time drift);
• Pure skewness: $L^x_t(X)$3 (asymmetric crossing);
• Pure stickiness: $L^x_t(X)$4 (symmetric slow passage with positive sojourn);
• Skew-sticky: $L^x_t(X)$5 (asymmetric and sticky, interpolating between skew and sticky cases).

The scale function and speed measure for such SDEs are explicitly computable: $L^x_t(X)$6 This structural representation aligns with classical diffusion theory and provides a unified platform for occupation, local time, and boundary analysis (Anagnostakis et al., 2024, Salminen et al., 2019).

In higher dimensions or on networks (e.g., star graph $L^x_t(X)$7), the sticky and skew-sticky boundary conditions generalize to

$L^x_t(X)$8

where $L^x_t(X)$9 is the edgewise generator, $x$0 are edge transmission weights, $x$1 is stickiness, and $x$2 is the vertex (Berry et al., 2024).

3. Time-change Constructions and Local Time Analysis

Time-change techniques are central in constructing sticky and sticky-skew diffusions. If $x$3 is the underlying (nonsticky) reflected or Walsh diffusion and $x$4 is its local time at the interface, the sticky process is

$x$5

such that the occupation time at the interface is proportional to the accumulated local time: $x$6 This mechanism directly translates stickiness into sojourns at the interface, with the accumulated local time enforcing the sticky threshold (Berry et al., 2024).

For skewness, the singular drift term in the SDE acts as a pushing force in the direction of the interface, modulated by the sign and magnitude of $x$7. The time-change and local time techniques generalize to multidimensional settings and to interfaces of vanishing thickness through stochastic homogenization, yielding limiting drift and diffusive corrections (Aryasova et al., 15 Dec 2025).

4. Existence, Uniqueness, and Pasting Theorems

Well-posedness (existence and uniqueness) of sticky-threshold and skew diffusions requires analysis of SDEs with singular, measure-valued drift, and possibly discontinuous diffusion. The general methodology relies on:

• Weak/strong existence per regime: Existence and uniqueness in law (weak) or pathwise uniqueness (strong) on each side of a threshold;
• Compatibility: Coincidence of SDE coefficients in neighborhoods of the threshold;
• Pasting theorem: Assembling global solutions from local components on either side of the threshold under compatibility (first strong explicit pasting result (Mazzonetto et al., 23 Apr 2026));
• Local time comparison: For purely sticky or threshold cases (no skewness), pathwise uniqueness can be verified through vanishing right local time of the difference of solutions.

In the presence of stickiness ($x$8), uniqueness is generally only weak, while for pure skewness strong uniqueness holds (Mazzonetto et al., 23 Apr 2026). These techniques extend to threshold CKLS/CIR-type diffusions, unifying previous models of reflecting, sticky, skew, and regime-switching interface SDEs.

5. Asymptotic and Homogenization Regimes

Asymptotic regimes arise naturally when interfacial parameters (e.g., thickness $x$9, skew parameter $\nu$0, killing rate $\nu$1) vary, yielding effective macroscopic boundary behaviors. Explicit characterization of:

• Neumann (reflecting): $\nu$2, or $\nu$3;
• Robin (sticky): $\nu$4, or $\nu$5;
• Dirichlet (absorbing): $\nu$6, or $\nu$7.

The sticky-threshold regime interpolates between pure reflection and absorption, with the Robin boundary condition acting as the analytic/probabilistic signature of stickiness (Capitanelli et al., 2014).

In stochastic homogenization, as the spacing between sticky or skew interfaces tends to zero and the local-time coefficients scale accordingly, the effective limiting SDE develops averaged drift and reduced diffusion coefficients, capturing the macroscopic influence of dense semipermeable and sticky membranes (Aryasova et al., 15 Dec 2025).

6. Occupation Times, Moment Generating Functions, and Estimation Theory

Occupation time and local time at the interface underpin many theoretical and inferential results:

• Law of occupation time: The MGF and moments of time spent above/below the threshold can be computed via the Green kernel and recursive moment equations, accommodating skew and sticky modifications (Salminen et al., 2019): $\nu$8 Closed-form expressions are available for skew Brownian and sticky Brownian motion.
• Statistics of high-frequency data: Approximations of local time using path functionals support consistent estimation of the skewness and stickiness parameters from discrete observation data, even in the presence of sticky-skew-oscillating interfaces (Anagnostakis et al., 2024).
• Optimal stopping: The value function and location of optimal stopping thresholds for sticky and skew Brownian motion admit explicit equations based on the Riesz representation, revealing when smooth fit (SF) and scale-smooth fit (SSF) principles hold or fail at the threshold. The presence of atoms in the representing measure, as in sticky Brownian motion with sufficiently large discount parameter, results in the breakdown of SF/SSF (Crocce et al., 2013).

7. Generalizations and Physical Contexts

Sticky-threshold and skew diffusions extend to complex domains and higher-dimensional geometries:

• Prefractal boundaries: Skew Brownian motion across Koch snowflake boundaries, with analysis of asymptotic regimes recovers reflecting, sticky, and absorbing limits, supported by explicit Revuz-measure and additive functional calculus (Capitanelli et al., 2014).
• Star graphs and networks: Sticky diffusions on star graphs involve boundary conditions coupling edge-derivatives and stickiness at the vertex, accommodating arbitrary numbers of outgoing edges with explicit Itô/Freidlin–Sheu formulas (Berry et al., 2024).
• Physical and engineering applications: These process classes model anomalous interface transport, heterogeneous media, regime-switching, and interface-induced slowdowns observed in statistical physics and finance. Sticky and skew behavior naturally emerge in stochastic models with semipermeable barriers, jump-diffusions, and inhomogeneous materials (Aryasova et al., 15 Dec 2025).

Process Type	SDE Singularity	Interface Effect
Skew Brownian motion	Point mass in $\nu$9	Asymmetric crossing
Sticky Brownian motion	Atom in speed measure	Positive sojourn
Skew-sticky threshold SDE	Both	Both
Threshold regime switching	Jumps in $\nu(dx) = \beta\,\delta_\theta(dx)$0	Discontinuous coeff.

These structural features can be flexibly combined and parameterized, enabling a broad spectrum of diffusive interface phenomena within a rigorous probabilistic and analytic framework (Berry et al., 2024, Mazzonetto et al., 23 Apr 2026, Aryasova et al., 15 Dec 2025).