Feller's Brownian Motion & Boundary Behaviors

• Feller's Brownian Motion is a strong Markov process class that generalizes Brownian motion by incorporating explicit boundary conditions.
• It utilizes the Feller semigroup approach to define generator domains that encode absorbing, reflecting, sticky, elastic, and boundary-jump behaviors.
• The framework links continuous diffusions with discrete birth–death processes, enabling precise modeling of complex boundary phenomena.

Feller's Brownian Motion refers to a canonical class of strong Markov processes that generalize one-dimensional Brownian motion by incorporating a comprehensive set of boundary behaviors at specific points (often at the origin). This family is rigorously defined via the Feller semigroup approach and characterized by explicit generator domains encoding absorption, reflection, stickiness, and boundary jumps. Feller's theory allows both spatially homogeneous and inhomogeneous processes, providing a unifying formalism for diffusions with arbitrary local behaviors at the boundary.

1. Feller Processes and Semigroups

A Feller process is a strong Markov process whose transition semigroup $(P_t)_{t\ge0}$ is a strongly continuous contraction semigroup on $C_0(E)$, the space of continuous functions vanishing at infinity on the state space $E$ (Böttcher, 2010, Kostrykin et al., 2010). The infinitesimal generator $A$ is defined for such semigroups by

$$D(A) = \left\{ f\in C_0(E): \lim_{t\to0}\frac{P_t f - f}{t} \text{ exists in } \|\cdot\|_\infty \right\}, \qquad Af = \lim_{t\to0}\frac{P_t f - f}{t}.$$

Brownian motion on $\mathbb{R}^d$ is a prototypical Feller process. For the half-line $[0,\infty)$ or finite intervals, Feller's work provides a complete analytic description of all possible Feller processes generated by $A f(x) = \frac{1}{2} f''(x)$ on the interior, with the domain determined by “Wentzell boundary conditions” (Kostrykin et al., 2010).

2. General Boundary Conditions: The Feller Quadruple

On $E = [0,\infty)$ with cemetery $\Delta$, the most general Feller's Brownian motion (FBM) is specified by a quadruple $(p_1,p_2,p_3,p_4)$, where $p_1, p_2, p_3 \ge 0$ and $p_4$ is a $\sigma$-finite measure on $(0,\infty)$ satisfying $\int (x \wedge 1) p_4(dx) < \infty$ (Li et al., 31 Dec 2025, Li, 2024, Erhard et al., 21 Dec 2025). The generator acts as

$L f(x) = \tfrac{1}{2} f''(x), \quad x > 0,$

with domain

$$D(L) = \left\{ f\in C_0([0,\infty)) \cap C^2((0,\infty)) : f'' \in C_0([0,\infty)),\, p_1 f(0) - p_2 f'(0) + \tfrac{p_3}{2} f''(0) + \int_{(0,\infty)} [f(0)-f(x)] p_4(dx) = 0 \right\}.$$

This boundary condition encodes:

• Absorbing: $p_2 = p_3 = p_4 = 0$, $p_1 > 0$, i.e., $f(0) = 0$.
• Reflecting: $p_1 = p_3 = p_4 = 0$, $p_2 > 0$, i.e., $f'(0) = 0$.
• Elastic: $p_3 = p_4 = 0$, $p_1, p_2 > 0$, i.e., $p_1 f(0) = p_2 f'(0)$.
• Sticky: $p_1 = 0$, $p_2, p_3 > 0$, $p_4 = 0$, i.e., $-p_2 f'(0) + \frac{p_3}{2} f''(0) = 0$.
• Boundary-jump: $p_4 \neq 0$, allows instantaneous jumps from 0 into the interior.

The quadruple is unique up to an overall scaling (Li et al., 31 Dec 2025, Li, 2024).

3. Excursion and Local Time Constructions

Feller's construction leverages Itô's excursion theory (Erhard et al., 21 Dec 2025, Li et al., 31 Dec 2025). For the general case, the trajectory is glued from standard Brownian excursions away from zero, combined with independent mechanisms at the boundary:

• Killing clock (rate $\lambda = p_1$) sends the process to the cemetery.
• Reflection ($p = p_2$) yields instantaneous departure along a new excursion.
• Stickiness ($\kappa = p_3/2$) causes sojourns at zero proportioned to local time.
• Jump-in ($p_4$) restarts at an interior point determined by $p_4$ after each boundary hit.

This yields strong Markov processes with continuous sample paths (off killing), whose excursion measures on $(0,\infty)$ agree with standard Brownian motion. For more general inhomogeneous diffusions, such as Feller-type Brownian motion with state-dependent coefficients, the process is built by mixing state-dependent Lévy processes, with local behavior governed by a family of characteristic exponents (Böttcher, 2010).

4. Resolvents, Semigroups, and Explicit Formulas

The resolvent operator $R_\alpha = (\alpha - L)^{-1}$ can be written in terms of the classical killed Brownian resolvent and boundary-adjusting terms. For $h\in C_0([0,\infty))$, (Li et al., 31 Dec 2025): $$G_\alpha h(x) = G^0_\alpha h(x) + G_\alpha h(0) e^{-\sqrt{2\alpha} x},$$ where $G^0_\alpha$ is the resolvent for Brownian motion killed at zero, and

$$G_\alpha h(0) = \frac{2 p_2 \int_0^\infty e^{-\sqrt{2\alpha} y} h(y) dy + p_3 h(0) + \int_0^\infty G^0_\alpha h(y) p_4(dy)}{p_1 + \sqrt{2\alpha} p_2 + \alpha p_3 + \int_0^\infty (1-e^{-\sqrt{2\alpha} y}) p_4(dy)}.$$

Transition densities and Green's functions for all possible boundary behaviors (absorbing, reflecting, sticky, elastic, and generalized Wentzell) have been constructed in closed form (Kostrykin et al., 2010).

5. Pathwise Realizations and Invariance Principles

Explicit sample path constructions with local-time mechanisms are available for all parameter regimes (Kostrykin et al., 2010, Li, 2024). For example:

• Sticky Reflection: Time-change of reflected Brownian motion by $\tau^{-1}(t)=t+\kappa L_t$ yields sticky motion; the corresponding generator satisfies $\tfrac{\kappa}{2} f''(0) = f'(0)$ at the boundary.
• Elastic: Reflecting Brownian motion is killed at a local-time dependent exponential time.
• General: Each visit to 0 triggers holding (stickiness), possible jump-in, or killing, according to the quadruple parameters.

A discrete-to-continuum invariance principle holds: for any FBM, there exists a sequence of boundary random walks on $\mathbb{N}_0$ whose rescaled laws converge to the FBM limit under the Skorokhod topology (Li et al., 31 Dec 2025).

6. Extensions and Classification

Feller's original classification (regular, exit, entrance, natural) is determined by the corresponding scale and speed measures for one-dimensional diffusions (Kostrykin et al., 2010). Modern treatments also include processes on the real line with skew, sticky, and killing at zero (“Skew Sticky Brownian Motion Killed at Zero”), and on the union of two half-lines with “snapping-out” at the origin (Erhard et al., 21 Dec 2025). The generator domains are specified by linear conditions on $f(0), f'(0\pm), f''(0\pm)$ (with suitable matching or jumping terms) and excursion-gluing constructions.

7. Connections with Birth–Death Processes

Every Feller’s Brownian motion with a general boundary quadruple can be transformed into a continuous-time birth–death process via a suitable time change (Li, 2024). The boundary behaviors at zero correspond to absorption, instantaneous reflection, and entry for the birth–death chain at infinity, with the sticky sojourn at 0 (mass in the speed measure) “killed” by the time-change and not appearing in birth–death transition rates. This mapping provides a deep connection between continuum diffusions and discrete Markov chains.

---

References:

• (Böttcher, 2010) Feller Processes: The Next Generation in Modeling. Brownian Motion, Lévy Processes and Beyond
• (Kostrykin et al., 2010) Brownian Motions on Metric Graphs: Feller Brownian Motions on Intervals Revisited
• (Erhard et al., 21 Dec 2025) The Most General Brownian Motion on the Line and on Two Closed Half-Lines
• (Li et al., 31 Dec 2025) From boundary random walks to Feller's Brownian Motions
• (Li, 2024) Time-changed Feller's Brownian motions are birth-death processes