Skew Brownian motion: A model for diffusion with interfaces?

Skew Brownian motion is a diffusion process on the real line with a distinguished point. A particle diffusing under skew Brownian motion behaves as if it were experiencing ordinary Brownian motion except at the distinguished point, but at the distinguished point it moves to the left with a probability P or to the right with probability 1 - P, with P not equal to 1/2. As such, skew Brownian motion may be a reasonable model for diffusion with interfaces. An application is given to an ecological model with two types of habitat. However, if skew Brownian motion is formulated in terms of diffusion equations, assuming conservation of mass leads to predictions that in some cases a nonzero number of individuals must be at the distinguished point; i.e. that the probability distribution for the position of a particle may include a delta function at the distinguished point. In the ecological context there is some empirical work suggesting that individuals might in fact sometimes congregate on interfaces. It is unclear whether this behavior is problematic in other modeling contexts. The process of skew Brownian motion should be studied further to assess its usefulness in modeling diffusion in the presence of interfaces.