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On the constructions of the skew Brownian motion. (English) Zbl 1189.60145
This article is a well-written and in-depth survey of skew Brownian motion. Skew Brownian motion behaves as a scalar Brownian motion away from the origin, but the signs of its excursions from the origin are independent Bernoulli random variables allowing for a general probability $$p$$ that an excursion is positive. (When $$p=1/2$$, one of course obtains Brownian motion.) The survey presents all the various ways of constructing and characterizing skew Brownian motion and proves their mutual equivalence. These approaches are as follows: semigroup theory using an appropriate infinitesimal generator with singular (distribution-valued) coefficients; Dirichlet form theory; characterization by scale function and speed measure; characterization as the solution of a stochastic differential equation with a local time term; a martingale problem approach; construction using excursion theory; and derivation of skew Brownian motion as a limit of appropriately scaled random walks. The paper also describes basic properties of the flows induced by skew Brownian motion and extensions in which the stochastic differential equation for skew Brownian motion is generalized to allow spatial variation in the local drift and diffusion coefficients and variable probabilities for positive excursions, as well as multi-dimensional extensions. These generalizations lead to a consideration of some diffusion models on graphs. Finally, there is a brief survey of literature on applications of skew Brownian motion.

##### MSC:
 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J55 Local time and additive functionals 60J65 Brownian motion 60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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