In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations. The condition is named after the Swedish mathematician Lars Hörmander.
Definition 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
[ V , W ] ( x ) = D V ( x ) W ( x ) − D W ( x ) V ( x ) , {\displaystyle [V,W](x)=\mathrm {D} V(x)W(x)-\mathrm {D} W(x)V(x),}
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
A j 0 ( x ) , [ A j 0 ( x ) , A j 1 ( x ) ] , [ [ A j 0 ( x ) , A j 1 ( x ) ] , A j 2 ( x ) ] , ⋮ 0 ≤ j 0 , j 1 , … , j n ≤ n {\displaystyle {\begin{aligned}&A_{j_{0}}(x)~,\\&[A_{j_{0}}(x),A_{j_{1}}(x)]~,\\&[[A_{j_{0}}(x),A_{j_{1}}(x)],A_{j_{2}}(x)]~,\\&\quad \vdots \quad \end{aligned}}\qquad 0\leq j_{0},j_{1},\ldots ,j_{n}\leq n}
span Rd. They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index j 0 {\displaystyle j_{0}} taking only values in 1,...,n.
Application to stochastic differential equations Consider the stochastic differential equation (SDE)
d x = A 0 ( x ) d t + ∑ i = 1 n A i ( x ) ∘ d W i {\displaystyle \operatorname {d} x=A_{0}(x)\operatorname {d} t+\sum _{i=1}^{n}A_{i}(x)\circ \operatorname {d} W_{i}}
where the vectors fields A 0 , … , A n {\displaystyle A_{0},\dotsc ,A_{n}} are assumed to have bounded derivative, ( W 1 , … , W n ) {\displaystyle (W_{1},\dotsc ,W_{n})} the normalized n-dimensional Brownian motion and ∘ d {\displaystyle \circ \operatorname {d} } stands for the Stratonovich integral interpretation of the SDE. 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.
Application to the Cauchy problem With the same notation as above, define a second-order differential operator F by
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