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We construct diffusions with values in the nonnegative orthant, normalreflection along each of the axes, and two pairs of local drift/variancecharacteristics assigned according to rank; one of the variances is allowed tovanish, but not both. The construction involves solving a system of coupledSkorokhod reflection equations, then ``unfolding'' the Skorokhod reflection ofa suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79(2009) 534-536). Questions of pathwise uniqueness and strength are alsoaddressed, for systems of stochastic differential equations with reflectionthat realize these diffusions. When the variance of the laggard is at least aslarge as that of the leader, it is shown that the corner of the quadrant isnever visited.

Diffusions with rank-based characteristics and values in the nonnegative quadrant