Hörmander's condition
In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and . The condition is named after the Swedish mathematician Lars Hörmander.
Given two C1 vector fields V and W on d-dimensional Euclidean space Rd, let [V, W] denote their Lie bracket, another vector field defined by
where DV(x) denotes the Fréchet derivative of V at x ∈ Rd, which can be thought of as a matrix that is applied to the vector W(x), and vice versa.
Let A0, A1, ... An be vector fields on Rd. They are said to satisfy Hörmander's condition if, for every point x ∈ Rd, the vectors
span Rd. They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index taking only values in 1,...,n.
Now consider the stochastic differential equation
where the vectors fields are assumed to have bounded derivative. Hörmander's theorem asserts that if the SDE above satisfies the parabolic Hörmander condition, then its solutions admit a smooth density with respect to Lebesgue measure.
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