 White-noise vertical shear gives an Ornstein-Uhlenbeck inclination

ID: white-noise-vertical-shear-gives-an-ornstein-uhlenbeck-inclination

In the long-time growing horizontal-gradient branch, independent vertical shear noises combine as

cos ϕ d W_{1} + sin ϕ d W_{2} = d W

. The signed inclination

β = k_{3} e^{- c t} / k_{0}

c > 0

its stationary density is

c / (π g^{2}) e^{- c B^{2} / g^{2}}

g^{2} / (2 c)

. The Fokker-Planck probability current vanishes at infinity and gives this Gaussian by one integration. At

c = 0

and nonzero noise there is no normalizable stationary density.

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