Skew Brownian Motion: One Biased Site and the Asymmetry It Buys

Skew Brownian motion is ordinary Brownian motion everywhere except one point: at the origin each excursion chooses the positive side with probability $\alpha$ and the negative side with probability $1-\alpha$. Nothing else is altered — no drift, no varying diffusivity, no boundary. This note sets out how completely that single site fixes the law. The transition density from the origin is the normal density doubled and split, $2\alpha\varphi_t$ to the right and $2(1-\alpha)\varphi_t$ to the left, so the two halves keep the Brownian shape and differ only in the mass they carry. The probability of being positive is exactly $\alpha$ at every time, the expected occupation of the positive half-line is $\alpha t$ with no transient, and the mean displacement is $(2\alpha-1)\sqrt{2t/\pi}$.

What the parameter does not touch is as sharp as what it does. Every even moment equals that of ordinary Brownian motion, for every $\alpha$: the interface moves mass from one side to the other without changing how far the particle travels. An experiment that reports only a width or a dispersion coefficient carries no information about the interface at all.

A lattice walk that is fair at every site but the origin recovers all of it. At $\alpha = 0.75$ it returns the probability of being positive as $0.75023 \pm 0.00098$, the mean as $0.39858 \pm 0.00205$ against $0.398942$, the second moment as $1.00067 \pm 0.00317$ against $1$, and the occupation as $0.74995 \pm 0.00068$ — each within a quarter of a standard error. The one genuine difference between lattice and continuum is the atom the walk places at the origin: it is of order the lattice spacing, it belongs to neither side, and apportioning it by $\alpha$ is a derived correction that tracks across four halvings of the spacing.