Skew Sticky Brownian Motion Killed at Zero

• Skew sticky Brownian motion killed at zero is defined as a one-dimensional Markov process incorporating asymmetric reflection, positive holding time, and probabilistic killing at the origin.
• Its construction employs a stochastic differential equation using local time at zero and a boundary condition that unifies classical reflecting, absorbing, and elastic Brownian motions.
• Applications include scaling limits of random walks, diffusion across interfaces, and models in finance and engineering, with open research on multidimensional generalizations and parameter estimation.

A skew sticky Brownian motion killed at zero (SSBMK) is a one-dimensional strong Markov process on $\R_\Delta = \R \cup \{\Delta\}$, where $\Delta$ denotes a cemetery state. The process behaves as standard Brownian motion away from the origin, but exhibits a combination of three boundary phenomena at $0$: asymmetric reflection ("skewness"), positive holding time ("stickiness"), and probabilistic absorption into the cemetery ("killing"). The SSBMK process is characterized by three parameters: $0 \le \beta \le 1$ (skewness), $c \ge 0$ (stickiness), and $\gamma \ge 0$ (killing rate), yielding the most general possible boundary behavior for one-dimensional Feller Brownian motions on the full real line. It unifies several classical processes as special cases, including pure, skew, sticky, and elastic Brownian motion. The SSBMK is fundamental in the classification of Brownian motions with general boundary behavior at a point, as shown by Erhard–Franco–Muricy (Erhard et al., 21 Dec 2025).

1. Formal Construction and Stochastic Differential Description

The SSBMK process $(X_t)_{t \ge 0}$ is a càdlàg strong Markov process with state space $\R_\Delta$, constructed as follows:

• For $t < T_0 = \inf\{s > 0 : X_s = 0\}$, $X_t$ evolves as standard Brownian motion.
• Upon hitting $0$, the process exhibits sticky, skew, and killing behavior:
  • "Skew": Upon leaving $0$, the process jumps to the right ($0^+$) with probability $\beta$ and to the left ($0^-$) with probability $1-\beta$.
  • "Sticky": The process spends a positive Lebesgue measure of time at $0$, with mean holding time parameter $c$.
  • "Killing": At $0$, the process may jump to the cemetery $\Delta$ at random local time, with rate $\gamma$.

This behavior is formalized via a stochastic differential equation involving the local time at $0$: $$dY_t = dB_t + (2\beta - 1) dL^0_t - \tfrac{1}{c} 1_{\{Y_t = 0\}} dt,$$ with killing to $\Delta$ the first time the accumulated local time $L^0_t$ exceeds an independent exponential random variable of parameter $\gamma$ (Erhard et al., 21 Dec 2025).

2. Infinitesimal Generator and Boundary Conditions

The SSBMK process is the unique Feller–Wentzell process with generator

$$Lf(x) = \begin{cases} \tfrac{1}{2} f''(x), & x \ne 0, \ 0, & x = \Delta, \end{cases}$$

with a domain consisting of continuous functions $f \in C(\R_\Delta)$, twice differentiable off $0$, whose second derivatives extend continuously to $0$ and which satisfy the boundary condition

$$c\,f''(0) = \beta f'(0^+) - (1-\beta) f'(0^-) - \gamma f(0),$$

for parameters $\beta, c, \gamma$ as above. This condition interpolates:

• Neumann (reflecting, for $c = \gamma = 0$),
• Dirichlet (absorbing, for $c = 0$, $\gamma \to \infty$),
• sticky and skew boundaries,
• and elastic (partly reflecting, partly killing) boundaries (Erhard et al., 21 Dec 2025).

3. Scale Function and Speed Measure

The process admits a natural scale function and speed measure:

• Scale function

$$s(x) = \begin{cases} x / (1-\beta), & x < 0, \ x / \beta, & x > 0 \end{cases}$$

which encodes the skewness of boundary exiting.

• Speed measure

$$m(dx) = 2(1-\beta) 1_{x<0} dx + 2\beta 1_{x>0} dx + 2c\,\delta_0(dx),$$

i.e., Lebesgue measure on positive/negative real line, weighted by exit probabilities, and an atomic component at $0$ proportional to $c$ (stickiness parameter) (Erhard et al., 21 Dec 2025).

4. Transition Semigroup and Resolvent

The resolvent $R_\lambda(x, dy)$ of the SSBMK, which is the Laplace transform of the transition semigroup, decomposes into an absolutely continuous part and an atom at the cemetery: $$R_\lambda(x, dy) = r_\lambda(x,y) dy + m_\lambda(x)\,\delta_\Delta(dy),$$ where for $x, y \neq 0$,

$$r_\lambda(x,y) = \frac{1}{\sqrt{2\lambda}}\left( e^{-\sqrt{2\lambda}|x - y|} + K(\lambda)\,e^{-\sqrt{2\lambda}(|x| + |y|)} \right),$$

and

$$m_\lambda(x) = \frac{\gamma + \sqrt{2\lambda}c}{\lambda + \gamma + \sqrt{2\lambda}c} e^{-\sqrt{2\lambda}|x|},$$

with $K(\lambda)$ as specified in (Erhard et al., 21 Dec 2025). The presence of $m_\lambda(x)$ reflects the possibility of killing at $0$.

5. Excursion Structure and Lévy System

The process away from $0$ follows standard Brownian excursions. The behavior at $0$ is described via the excursion law:

• Probability of departing to the right/left is governed by $\beta$.
• The local time inverse $T_\ell = \inf\{t: L^0_t > \ell\}$ has jump rate $\gamma$ to the cemetery, holds at $0$ for mean time $c$, then resumes according to the skew law.
• The Lévy system kernel at $0$ is

$$N(0, dy) = \gamma\,\delta_\Delta(dy) + \tfrac1c \left[\beta\,\delta_{0^+} + (1-\beta)\,\delta_{0^-}\right](dy),$$

with no jumps for $x \neq 0$.

6. Special Cases

The SSBMK encompasses several classical processes as degenerate or limiting cases:

Process Type	Parameters	Boundary Behavior
Standard BM	$\beta=1/2$, $c=0$, $\gamma=0$	Pure reflection
Skew BM	$\beta\ne1/2$, $c=0$, $\gamma=0$	Asymmetric reflection
Sticky BM	$\beta=1/2$, $c>0$, $\gamma=0$	Positive holding at $0$
Elastic BM	$\beta=1/2$, $c=0$, $\gamma>0$	Partial killing
SSBMK (general)	any $\beta$, $c$, $\gamma\ne0$	All combined

Varying $\beta$ modifies the asymmetry at $0$, $c$ modulates the stickiness and holding time, and $\gamma$ sets the rate of absorption at $0$.

7. Applications and Open Problems

Processes of SSBMK type arise as scaling limits of random walks or particle systems with inhomogeneous jump behavior at a special point, e.g., the "slow bond" random walk. Physical models of diffusion with interface permeability, surface adhesion, or trapping employ SSBMK to represent partial reflection, trapping, or absorption at interfaces. SSBMK also appears in financial mathematics and engineering models for transport across semi-permeable membranes.

Research directions include:

• Extension to multidimensional analogues with sticky/skew/killing interfaces,
• Statistical estimation of $(\beta, c, \gamma)$ from discretely observed data,
• Large-time occupation asymptotics near sticky interfaces.

The classification result of (Erhard et al., 21 Dec 2025) shows that no other boundary behaviors are possible for strong Markov Brownian motions on the line—any such process is either a SSBMK or a mixture/specialization thereof. Explicit formulas and proofs are described in Borodin & Salminen (Section 13, Appendix I) and in Erhard–Franco–Muricy (2024) (Erhard et al., 21 Dec 2025).