Skew Sticky Killed at Zero Snapping Out BM

• SSKSOBM is a unified model that generalizes skew, sticky, killed, and snapping-out boundary behaviors for one-dimensional diffusions.
• The framework employs Feller–Wentzell boundary conditions to derive explicit generators, excursion-theoretic constructions, and resolvent kernel formulas.
• A discrete approximation via nearest-neighbor random walks is presented, converging to the continuous SSKSOBM and ensuring robust mathematical validation.

The Skew Sticky Killed at Zero Snapping Out Brownian Motion (SSKSOBM) is the most general strong Markov process whose excursions away from the two-sided singularity at zero coincide with those of standard Brownian motion, with the option to be sent to a cemetery state upon hitting either $0^-$ or $0^+$. The state space is

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

where $0^-$ and $0^+$ are two distinct "zeros" and $\Delta$ is a cemetery point. The SSKSOBM unifies and extends classical skew, sticky, killed, and snapping-out boundary behaviors, providing a unified Feller–Wentzell boundary description for the full spectrum of regular one-dimensional diffusions on $G_\Delta$ (Erhard et al., 21 Dec 2025).

1. Generator and Feller–Wentzell Boundary Conditions

The infinitesimal generator $\mathcal L$ of SSKSOBM is given by

$$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$

with domain $\mathcal D(\mathcal L) \subset C(G_\Delta)$ consisting of functions $0^+$0 such that $0^+$1 exists on $0^+$2 extending continuously to $0^+$3, and satisfying the following Feller–Wentzell boundary conditions at $0^+$4 and $0^+$5:

$0^+$6

subject to

$0^+$7

Here, $0^+$8 is the killing rate at $0^+$9 (jump to $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$0), $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$1 is the snapping-out rate (jump to the opposite zero), $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$2 encodes reflection (elastic) at $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$3, and $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$4 is the stickiness parameter.

2. Excursion-Theoretic Construction

Let $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$5 denote the continuous local time of the process at $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$6. Away from zero (on each half-line), the process behaves as standard Brownian motion. Upon a return to $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$7 (similarly, $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$8), one of four events occurs:

• With probability $$G_\Delta = (-\infty,0^-] \cup [0^+,\infty) \cup \{\Delta\}$$9, the process is killed and sent to $0^-$0,
• With probability $0^-$1, the process "snaps out" to $0^-$2, immediately starting a new Brownian excursion from $0^-$3,
• With probability $0^-$4, the process reflects back into $0^-$5,
• With probability $0^-$6, the process remains "stuck" at $0^-$7 for an exponential holding time, then continues off $0^-$8 as Brownian motion.

The transition events at each local-time increase are realized via four independent Poisson processes on $0^-$9 with respective intensities proportional to $0^+$0, and a symmetric construction applies at $0^+$1 (Erhard et al., 21 Dec 2025).

3. Scale, Speed, and Jump Measures

On each half-line, the scale function is linear: $0^+$2 for $0^+$3 or $0^+$4. The speed measure is $0^+$5, and stickiness at $0^+$6 introduces an atom at $0^+$7 to the speed measure: $0^+$8, so that sojourn times at $0^+$9 are slowed by a factor $\Delta$0.

The snapping-out mechanism yields a purely atomic jump measure:

$\Delta$1

so that from $\Delta$2 (resp. $\Delta$3) there is a rate $\Delta$4 (resp. $\Delta$5) jump to $\Delta$6 (resp. $\Delta$7).

Parameter	Interpretation	Constraint
$\Delta$8	Killing rate at $\Delta$9	$G_\Delta$0
$G_\Delta$1	Snapping-out rate	$G_\Delta$2
$G_\Delta$3	Reflection at $G_\Delta$4	$G_\Delta$5
$G_\Delta$6	Stickiness at $G_\Delta$7	$G_\Delta$8; at least one $G_\Delta$9

4. Resolvent Kernel, Laplace Transforms, and Hitting Times

Let $\mathcal L$0,

$\mathcal L$1

The $\mathcal L$2-resolvent density $\mathcal L$3 with respect to $\mathcal L$4 is expressed as a combination of $\mathcal L$5 and image terms enforcing the four boundary effects. On the right half-line ($\mathcal L$6),

$\mathcal L$7

where

$\mathcal L$8

is the effective reflection coefficient at $\mathcal L$9. Mixed-region terms (crossing between half-lines) appear proportionally to $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$0. Analogue formulas hold on the left.

The Laplace transform of the first hitting time of a boundary point $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$1 is retrieved as

$$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$2

This yields explicit, closed-form Laplace transforms for hitting times of $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$3 under general boundary behavior (Erhard et al., 21 Dec 2025).

5. Existence, Uniqueness, and Approximations

The Feller–Wentzell boundary system (FW), coupled with $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$4, uniquely determines a closed dissipative operator on $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$5. By the Hille–Yosida theorem, this is the generator of a Feller semigroup and thus a strong Markov process.

The well-posedness of the martingale problem is verified on the dense core

$$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$6

A discrete approximation is achieved via nearest-neighbor random walks on $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$7, with jump rates at $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$8 scaled as follows:

• Usual jumps in the bulk: $$\mathcal L f(x) = \frac{1}{2}f''(x), \quad x \in G, \qquad \mathcal Lf(\Delta) = 0,$$9,
• Killing at $\mathcal D(\mathcal L) \subset C(G_\Delta)$0: $\mathcal D(\mathcal L) \subset C(G_\Delta)$1,
• Snapping-out: $\mathcal D(\mathcal L) \subset C(G_\Delta)$2.

The random walks converge in distribution, as $\mathcal D(\mathcal L) \subset C(G_\Delta)$3, to the SSKSOBM with parameters $\mathcal D(\mathcal L) \subset C(G_\Delta)$4, $\mathcal D(\mathcal L) \subset C(G_\Delta)$5, $\mathcal D(\mathcal L) \subset C(G_\Delta)$6, $\mathcal D(\mathcal L) \subset C(G_\Delta)$7 after normalization (Erhard et al., 21 Dec 2025).

6. Relation to and Unification of Earlier Models

SSKSOBM extends and unifies several previously studied boundary phenomena:

• Classical skew-sticky-killed BM on $\mathcal D(\mathcal L) \subset C(G_\Delta)$8 is recovered by identifying $\mathcal D(\mathcal L) \subset C(G_\Delta)$9 (no snapping, $0^+$00), with $0^+$01 (killing), $0^+$02 (skewness), $0^+$03 (stickiness).
• Lejay's Snapping-Out Brownian Motion (SNOB) appears for $0^+$04, $0^+$05, $0^+$06, $0^+$07 (pure snapping without killing or stickiness).
• When all four parameters are present, SSKSOBM realizes new boundary dynamics in which skewness, snapping, stickiness, and killing are present in competition.

This framework provides the complete classification of strong Markov processes on $0^+$08 whose excursions away from zero are Brownian (Erhard et al., 21 Dec 2025).