Skew Sticky Killed at Zero Snapping Brownian Motion

• SSKSOBM is a comprehensive one-dimensional diffusion process that extends standard Brownian motion with skewness, stickiness, killing, and snap-out effects at the origin.
• It employs a probabilistic construction with explicit boundary parameters (c1, c2, c3, a) to encode interface behaviors such as reflection, absorption, sticky holding, and directional change.
• The model unifies various interface diffusion frameworks, offering analytical tools for studying flux discontinuities with applications in physics, imaging, and chemical transport.

The Skew Sticky Killed at Zero Snapping Out Brownian Motion (SSKSOBM) is the most general diffusion process on the union of two closed half-lines whose excursions from zero agree with those of standard Brownian motion, but whose behavior at zero simultaneously admits skewness, stickiness, killing, instantaneous reflection, and “snapping out” across the origin. This process generalizes the classic Snapping Out Brownian Motion (SOBM) of Lejay and encompasses a full family of boundary Markovian mechanisms characterized by four nonnegative parameters per side, providing a unified framework for interface diffusions exhibiting a combination of flux discontinuities, sticky sojourns, possible absorption, and direction change at the interface (Erhard et al., 21 Dec 2025, Lejay, 2016).

1. Probabilistic Construction and State Space

The SSKSOBM evolves as standard one-dimensional Brownian motion (BM) away from the origin, with non-trivial dynamics localized at zero. The state space is

$$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\},$$

where $\Delta$ denotes the absorbing cemetery state. Upon hitting $0^+$ (from the right) or $0^-$ (from the left), the process experiences a Markovian “boundary event” dictated by the quadruple of nonnegative parameters $(c_1^\pm, c_2^\pm, c_3^\pm, a^\pm)$ (all summing to 1):

• $c_1^\pm$: Killing at $0^\pm$ (probability of immediate jump to $\Delta$)
• $c_2^\pm$: Instantaneous reflection into the same half-line
• $c_3^\pm$: Sticky holding time at $0^\pm$, after which a further event occurs
• $a^\pm$: Snapping out across the origin to $0^\mp$ to begin a fresh excursion on the opposite half-line

When $X_t$ hits $0^+$ (or $0^-$), if $c_3^+ > 0$, the process remains at $0^+$ for an exponentially distributed holding time $\mathrm{Exp}(1/c_3^+)$ (“sticky” sojourn); at the end, one of the four post-holding transitions occurs, chosen in proportion to $(c_1^+, c_2^+, c_3^+, a^+)$ as above. The process on $0^-$ is symmetric with respect to the parameters $(c_1^-, c_2^-, c_3^-, a^-)$. In degenerate cases ($c_3^\pm=0$), the sticky holding time vanishes, yielding instantaneous transitions.

This construction generalizes the Lejay SOBM (recovered for $c_1^\pm=0, c_3^\pm=0, c_2^\pm=1$), piecing out Brownian motions on the two half-lines with Markovian interface rules (Erhard et al., 21 Dec 2025, Lejay, 2016).

2. Generator, Domain, and Boundary Value Characterization

On the interior of $G_\Delta\setminus\{0^\pm, \Delta\}$, the infinitesimal generator is the Laplacian

$$L f(x) = \tfrac12 f''(x), \qquad x \in (-\infty, 0^-) \cup (0^+, \infty),$$

with $L f(\Delta) = 0$. The domain $D(L)$ consists of those $f \in C(G_\Delta)$ which are $C^2$ away from the boundary points and for which $f''$ extends to $0^\pm$, subject to the following coupled boundary equations: \begin{align*} & c_1^+ f(0^+) + a^+ \left[f(0^+) - f(0^-)\right] - c_2^+ f'(0^+) + \tfrac12 c_3^+ f''(0^+) = 0, \ & c_1^- f(0^-) + a^- \left[f(0^-) - f(0^+)\right] + c_2^- f'(0^-) + \tfrac12 c_3^- f''(0^-) = 0. \end{align*} These relations subsume the classical interface boundary conditions (Dirichlet, Neumann, Robin/third-kind, skew, sticky), allowing robust interpolation and limiting transitions between them. The choice of parameters also recovers the semi-permeable barrier condition (Robin-jump type) associated with the SOBM (Erhard et al., 21 Dec 2025, Lejay, 2016).

3. Scale Function, Speed Measure, and Interface Parameters

The process admits canonical scale and speed structures:

• On each open half-line, the scale function is $s(x) = x$.
• The speed measure combines Lebesgue on each half-line with atomic masses at the boundaries:

$$m(dx) = 2\,dx \;\; (x \ne 0^\pm), \quad m(\{0^+\}) = c_3^+, \;\; m(\{0^-\}) = c_3^-.$$

The continuous part is invariant under changes to the skewness ($a^\pm$), killing ($c_1^\pm$), and reflection ($c_2^\pm$) parameters, which impact only boundary flux; stickiness ($c_3^\pm$) introduces persistent mass at the interface points, producing a singular correction to the speed measure (Erhard et al., 21 Dec 2025).

4. Resolvent Kernel and Transition Density

The resolvent kernel $R_\alpha(x, y)$, corresponding to $(\alpha - L)^{-1}$, satisfies the elliptic equation

$$(\alpha - \tfrac12 d^2/dx^2)R_\alpha(x, y) = \delta_y(x)$$

subject to the boundary equations above for $x=0^\pm$. For $x, y > 0$ (and analogously $x, y < 0$),

$$R_\alpha(x, y) = \frac{1}{\sqrt{2\alpha}} \left( e^{-\sqrt{2\alpha}|x-y|} + M_\alpha\, e^{-\sqrt{2\alpha}(x+y)} \right),$$

where $M_\alpha$ is determined by the specific boundary data. Transition kernels across the origin ($x > 0 > y$, $x < 0 < y$) are constructed by matching through the “snap-out” term ($a^\pm$) at $0^\pm$.

The transition density $p(t, x, y)$ is obtained by Laplace inversion:

$$\int_0^\infty e^{-\alpha t}\,p(t, x, y)\,dt = R_\alpha(x, y).$$

Closed-form formulas involve complementary Gaussian integrals modulated by the sticky, killing, and skew parameters. For classical cases, such as Lejay’s symmetric SOBM, explicit series and integral representations for $p(t, x, y)$ are available, revealing boundary layer singularities and jump-discontinuities at $y \to 0^\pm$ (Erhard et al., 21 Dec 2025, Lejay, 2016).

5. Excursion Law and Snapping-Out Modification

The SSKSOBM admits an excursion-theoretic description: local excursions away from $0^\pm$ are governed by the standard Itô measures $n^\pm$, but the choice of excursion on each interface crossing depends stochastically on the quadruple parameters. At each visit to $0^+$, an excursion starts from $n^+$ with probability $c_2^+ + c_3^+$ (reflection or sticky), or is “snapped out” to $n^-$ with chance $a^+$; killing at $0^+$ (with rate $c_1^+$) corresponds to absorbing the process at $\Delta$. The process can be constructed by thinning a Poisson point process of excursions, with the law of the SSKSOBM emerging from the applied thinning at local-time increments at zero (Erhard et al., 21 Dec 2025).

6. Special and Limiting Cases

The SSKSOBM encapsulates a hierarchy of classical and modern interface diffusions:

• Pure Skew Sticky Snapping-Out: $c_1^\pm \to 0$ (no killing), yielding a skew sticky process alternating sides with probability $a^\pm$ and sticky holding with mass $c_3^\pm$.
• Skew Killed Snapping-Out: $c_3^\pm \to 0$ (no stickiness); at each hit, the process is either killed, instantly reflected, or snapped out.
• Classical SOBM: $c_1^\pm = 0,\; c_3^\pm = 0,\; c_2^\pm = 1$ (no stickiness or killing, pure reflection), yielding Lejay’s process where Brownian paths are pieced together by flipping sign or continuing with equal probability, and the interface satisfies a Robin-jump (semi-permeable) boundary condition (Erhard et al., 21 Dec 2025, Lejay, 2016).

These limits demonstrate that varying $(c_1^\pm, c_2^\pm, c_3^\pm, a^\pm)$ interpolates among the full set of Feller–Wentzell interface diffusions, recovering familiar reflected, sticky, skew, killed, and snapping-out processes as special cases.

7. Connections, Applications, and Simulation

The SOBM and its generalizations appear in the analysis of interface problems involving semi-permeable barriers, flux discontinuities, and models of transport across thin membranes. Applications include scenarios in mathematical physics, brain imaging, and chemistry where diffusion with both sticky and jump-behavior at an interface is essential. The simulation of SSKSOBM for practical applications follows from schemes developed for the SOBM, utilizing Brownian bridge crossing tests and local-time-based snap/kill/stick mechanisms as described by Lejay (Lejay, 2016), with adaptation for the richer interface law of the full SSKSOBM.

The SSKSOBM, by encoding the most general strong Markov process on the union of two closed half-lines whose excursions off zero are standard BMs and which at zero may stick, snap, reflect, or be killed, effectively completes the Feller program for classifying one-dimensional diffusions with Markovian interfaces (Erhard et al., 21 Dec 2025).