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.
They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index
taking only values in 1,...,n. Consider the stochastic differential equation (SDE) where the vectors fields
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.
With the same notation as above, define a second-order differential operator F by An important problem in the theory of partial differential equations is to determine sufficient conditions on the vector fields Ai for the Cauchy problem to have a smooth fundamental solution, i.e. a real-valued function p (0, +∞) × R2d → R such that p(t, ·, ·) is smooth on R2d for each t and satisfies the Cauchy problem above.
The great achievement of Hörmander's 1967 paper was to show that a smooth fundamental solution exists under a considerably weaker assumption: the parabolic version of the condition that now bears his name.
be smooth vector fields on M. Assuming that these vector fields satisfy Hörmander's condition, then the control system is locally controllable in any time at every point of M. This is known as the Chow–Rashevskii theorem.