The Most General Brownian Motion on the Line and on Two Closed Half-Lines

TL;DR

This work fully generalizes Feller’s Brownian motion on the half-line to the real line and to the union of two closed half-lines, characterizing the most general Brownian motions via explicit generator-domain conditions at the origin. On the line, the limit process is the Skew Sticky Killed at Zero Brownian Motion; on two half-lines, the limit is a novel Skew Sticky Killed Snapping Out Brownian Motion that extends Lejay’s SNOB. The authors provide precise boundary-condition descriptions, establish the corresponding Feller generators, and prove functional CLTs showing convergence of boundary random walks to these limits, with simulations illustrating the phenomena. Collectively, the results unify and extend classical boundary-type Brownian motions (reflecting, sticky, skew, killed, SNOB) into a single, parameterized framework and highlight how boundary behavior governs diffusion limits in line and two-half-line geometries.

Abstract

In the 1950s, W. Feller characterized the most general Brownian motion on the closed half-line. He showed that any such process is a mixture of reflected, sticky, and killed Brownian motions. By most general Brownian motion, we mean a strong Markov process whose excursions away from zero coincide with those of standard Brownian motion, and which may be sent to the cemetery state upon hitting zero. In this work, we fully characterize the most general Brownian motion on the whole real line and on the union of two closed half-lines. Our results are twofold. First, we show that the most general Brownian motion on the line is the process known in the literature as the Skew Sticky Brownian Motion Killed at Zero (see Borodin and Salminen's book). Second, we prove that the most general Brownian motion on two closed half-lines is a process, which we call the Skew Sticky Killed at Zero Snapping Out Brownian Motion. This process extends the Snapping Out Brownian Motion introduced by A. Lejay in 2016.

Figures (5)

• Figure 1: Description of the general Brownian motion on the half-line according to the chosen values on the simplex $c_1+c_2+c_3=1$.
• Figure 2: Jump rates for the slow bond random walk
• Figure 3: The jump rates for the boundary random walk on ${\mathbb R}_{n, \Delta}$ are $n^2$ times the ones shown in the picture.
• Figure 4: The jump rates for the boundary random walk on ${\mathbb G}_{n, \Delta}$ are $n^2$ times the ones shown in the picture.
• Figure 5: Simulation of the Boundary Random Walk with parameters $A_+ = 0.25$, $A_- = 0.25$, $B_+ = 2$, $B_- = 2$, $C_+ = 6$, and $C_- = 4$. The macroscopic time is $t = 1$, the macroscopic initial position is $u = 1/3$ and the discrete parameter is $n = 500$. Note that the walk switches between half-lines and eventually goes to the cemetery. This is highlighted in red. Although not so evident from the picture, since $C_+> C_-$, the walk has a preference to stay in the negative half-line. The stickiness at the points $0+$ and $0-$ is visible.

Theorems & Definitions (41)

• Theorem 1.1: Feller, see Knight, Theorem 6.2, p. 157
• Definition 2.1
• Remark 1
• Theorem 2.2
• Remark 2
• Remark 3
• Remark 4
• Definition 2.3
• Theorem 2.4
• Remark 5