Hörmander's condition

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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[edit]

Given two C1 vector fields V and W on d-dimensional Euclidean space Rd, let [VW] denote their Lie bracket, another vector field defined by

[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

\begin{align}
&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{align}
\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 taking only values in 1,...,n.

Now consider the stochastic differential equation

dx = V_0(x) \; dt + \sum_{i=1}^m V_i(x) \circ dW_i

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.

Application to the Cauchy problem[edit]

With the same notation as above, define a second-order differential operator F by

F = \frac1{2} \sum_{i = 1}^{n} A_{i}^{2} + A_{0}.

An important problem in the theory of partial differential equations is to determine sufficient conditions on the vector fields Ai for the Cauchy problem

\begin{cases} \dfrac{\partial u}{\partial t} (t, x) = F u(t, x), & t > 0, x \in \mathbf{R}^{d}; \\ u(t, \cdot) \to f, & \mbox{ as } t \to 0; \end{cases}

has a smooth fundamental solution, i.e. a real-valued function p (0, +∞) × R2dR such that p(t, ·, ·) is smooth on R2d for each t and

u(t, x) = \int_{\mathbf{R}^{d}} p(t, x, y) f(y) \, \mathrm{d} y

satisfies the Cauchy problem above. It had been known for some time that a smooth solution exists in the elliptic case, in which

A_{i} = \sum_{j = 1}^{d} a_{ji} \frac{\partial}{\partial x_{j}},

and the matrix A = (aji), 1 ≤ j ≤ d, 1 ≤ i ≤ n is such that AA is everywhere an invertible matrix.

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.

See also[edit]

References[edit]