skew and reflected Brownian motion

Two of the simplest modifications of Brownian motion are built directly from local time, and both answer concrete questions: what if the process cannot go below 0 (reflected), and what if it is biased about which side of 0 it prefers on leaving (skew)? Both are strong Markov diffusions that are ordinary Brownian motion away from 0 but behave specially at the origin, and both are cleanest to define through Tanaka's formula and excursion theory.

Reflected Brownian motion is |B_t|, or equivalently the Skorokhod solution to dX = dW + dL, X >= 0, where L is the local time at 0 that pushes X up exactly enough to keep it nonnegative (it increases only when X = 0). Its generator is (1/2) d^2/dx^2 on (0, infinity) with a Neumann (reflecting) boundary condition at 0. By Tanaka, |B_t| = beta_t + L_t^0 with beta a Brownian motion, so reflected Brownian motion is a Brownian motion plus a singularly continuous boundary push. Skew Brownian motion with parameter p in [0,1] behaves like Brownian motion off 0 but, at each excursion away from 0, chooses to go positive with probability p and negative with probability 1 - p, independently across excursions. It solves the SDE dX = dW + (2p - 1) dL_t^0(X), where the drift acts only through local time at 0; p = 1/2 recovers ordinary Brownian motion, p = 1 gives reflected Brownian motion, and intermediate p gives an asymmetric crossing rule. Its transition density is the heat kernel with an image charge of weight (2p - 1). A related object, Walsh Brownian motion, sends excursions onto rays of a star at chosen angles.

Why it matters: reflected Brownian motion is the canonical model for queues at heavy traffic (the workload process), for storage and inventory, and for diffusions in domains with reflecting boundaries; the multidimensional version (Harrison-Reiman) is central to queueing networks. Skew Brownian motion models diffusion across an interface with a permeability mismatch (heat or solute crossing a membrane, or motion in media with discontinuous diffusivity). The honest subtleties: skew Brownian motion is a genuine strong solution of its SDE only because of deep results (Harrison-Shepp showed |2p - 1| <= 1 is required, and the solution is unique in law), and the local-time drift is singular — it is NOT a classical drift b(x) dt, so standard Lipschitz SDE theory does not apply. Reflected Brownian motion's L is not differentiable; the reflection is enforced by the singular local-time term, not by a smooth force.