Skew Sticky Brownian Motion

• Skew sticky Brownian motion is a strong Markov process defined by parameters for skewness, stickiness, and killing at zero, outlining its boundary interaction.
• It uses a time-change approach and Feller–Wentzell boundary conditions to achieve a precise formulation through excursion theory and local time analysis.
• Recent extensions incorporate non-local operators and integro-differential conditions, enhancing the framework to capture jump phenomena and sticky plateaus.

Skew sticky Brownian motion refers to the most general strong Markov process on the real line whose excursions away from a distinguished point (usually zero) coincide with those of standard Brownian motion, but which incorporates, at each visit to zero, mechanisms of skew reflection, sticky local holding, and exponential killing. This class fully characterizes all Brownian-type processes with zero as a regular boundary, including in both the classical and certain non-local extensions. The precise mathematical framework involves parameterizations by skewness, stickiness (local time-based sojourn), and killing rate, with boundary conditions formulated in terms of the generator’s domain and explicit excursion-theoretic constructions (Erhard et al., 21 Dec 2025, Colantoni, 2023).

1. Formal Definition and Parameterization

The skew sticky Brownian motion on $\mathbb{R}$ with killing at zero is uniquely determined by three real parameters:

• $\alpha \in [-1,1]$ (skewness)
• $\gamma \ge 0$ (stickiness)
• $\kappa \ge 0$ (killing rate at zero)

The process $X$ is a càdlàg strong Markov process on $\mathbb{R}\cup\{\Delta\}$, with $\Delta$ a cemetery state. Its construction is as follows:

• Until the first hitting time $\tau_0 = \inf\{t:X_t = 0\}$, $X$ is a standard Brownian motion absorbed at zero.
• At zero, three competitive mechanisms operate:
  • With instantaneous reflection, $X$ resumes at $\alpha \in [-1,1]$0 with probability $\alpha \in [-1,1]$1 and at $\alpha \in [-1,1]$2 with probability $\alpha \in [-1,1]$3.
  • With probability corresponding to an exponential random time with rate $\alpha \in [-1,1]$4, the process sticks at $\alpha \in [-1,1]$5, remaining there before resuming a new excursion.
  • Independently, the process may be sent irreversibly to the cemetery $\alpha \in [-1,1]$6 at rate $\alpha \in [-1,1]$7.

Excursion-wise, after each return to zero, the process remains stuck for an exponential$\alpha \in [-1,1]$8 holding time, after which it is killed with probability $\alpha \in [-1,1]$9 or continues with skewed reflection as described (Erhard et al., 21 Dec 2025).

2. Semimartingale Structure and Local Time

Let $\gamma \ge 0$0 denote the symmetric local time of $\gamma \ge 0$1 at zero. $\gamma \ge 0$2 may be constructed as a time-change of a killed skew Brownian motion $\gamma \ge 0$3: $\gamma \ge 0$4 where $\gamma \ge 0$5 is killed upon hitting zero, and the time change is determined by

$\gamma \ge 0$6

so that

$\gamma \ge 0$7

Pushing the Tanaka–Itô decomposition through the time change, the process admits the decomposition: $\gamma \ge 0$8 where $\gamma \ge 0$9 is a standard Brownian motion in the enlarged filtration. The stickiness parameter $\kappa \ge 0$0 explicitly modulates the time the process spends at zero, via the time change $\kappa \ge 0$1.

3. Generator, Boundary Conditions, and Domain

On $\kappa \ge 0$2, the infinitesimal generator is $\kappa \ge 0$3. At zero, the domain $\kappa \ge 0$4 is determined by the mixed Feller–Wentzell boundary condition: $\kappa \ge 0$5 with $\kappa \ge 0$6, $\kappa \ge 0$7. These constants can be reparameterized by: $\kappa \ge 0$8 yielding the classical boundary condition: $\kappa \ge 0$9 or equivalently,

$X$0

The admissible domain is

$X$1

(Erhard et al., 21 Dec 2025).

4. Scale Function, Speed Measure, and Resolvent

The process admits explicit scale and speed measures: $X$2 The atom $X$3 in the speed measure captures analytically the stickiness at $X$4. The resolvent (Green’s function) solves: $X$5 subject to the boundary conditions. The explicit solution for $X$6 is: $X$7 where

$X$8

The prefactors are validated by continuity and jump conditions at zero.

5. Transition Densities and Hitting Times

The transition kernel $X$9 is obtained via Laplace inversion of the resolvent, or by the method of images with an additional sticky term. For $\mathbb{R}\cup\{\Delta\}$0: $\mathbb{R}\cup\{\Delta\}$1 The Laplace transform of the hitting time $\mathbb{R}\cup\{\Delta\}$2 of zero, started from $\mathbb{R}\cup\{\Delta\}$3, is: $\mathbb{R}\cup\{\Delta\}$4 A nonzero $\mathbb{R}\cup\{\Delta\}$5 increases the expected hitting time, while $\mathbb{R}\cup\{\Delta\}$6 controls the directional bias in first return (Erhard et al., 21 Dec 2025).

6. Non-local and Analytic Generalizations

Recent work (Colantoni, 2023) broadens the framework: non-local skew and sticky Brownian motions are defined via non-local (Marchaud- and Caputo-type) operators at the origin, replacing boundary conditions by integro-differential conditions. These extensions include subordinators describing jumps or sticky plateaus at zero:

• Non-local skew Brownian motion exhibits (strong) Markov right-continuous paths with jumps at zero, sign determined by a skew coin and sizes from a Lévy measure.
• Non-local sticky Brownian motion is non-Markov, spending plateaus at zero induced by the inverse subordinator, with resumption determined by a coin toss.

Both types coincide with the classical skew sticky Brownian motion in the local limit (subordinator degenerates to drift, subordinators’ Laplace exponents $\mathbb{R}\cup\{\Delta\}$7).

7. Existence, Uniqueness, and Classification

A key result (Theorem 2.3, (Erhard et al., 21 Dec 2025)) shows: every strong Markov process on $\mathbb{R}\cup\{\Delta\}$8 whose excursions away from zero follow those of standard Brownian motion and which only experiences instantaneous rebirth, local sojourn, or killing at zero, is a skew sticky killed Brownian motion for some triple $\mathbb{R}\cup\{\Delta\}$9. The four constants in the Feller–Wentzell boundary classification suffice to exhaust all possible boundary behaviors at zero.

The process thus provides a complete classification of all Brownian-type motions with boundary interaction at a single point, both in terms of explicit pathwise construction and generator boundary analysis (Erhard et al., 21 Dec 2025).