Definition: A differential operator on , is called a diffusion operator if it can be written
where and are continuous functions on and if for every , the matrix is a symmetric and nonnegative matrix.
If for every the matrix is positive definite, then the operator is said to be elliptic. The first example of a diffusion operator is the Laplace operator on :
It is of course an elliptic operator.
One of the first property of diffusion operators is that they satisfy a maximum principle. Before we state this principle let us recall the following simple result from linear algebra.
Lemma. Let and be two symmetric and nonnegative matrices, then
Since is symmetric and non negative, there exists a symmetric and non negative matrix such that . We have then
The matrix is seen to be symmetric and nonnegative and thus has a non negative trace.
Proposition (Maximum principle for diffusion operators). Let be a smooth function that attains a local minimum at . If is a diffusion operator then .
and let be a smooth function that attains a local minimum at . We have
where is the symmetric and non negative matrix with coefficients and is the Hessian matrix of , that is the symmetric matrix with coefficients . Since is a local minimum of , is a non negative matrix. We can now use the previous lemma to get the expected result
It is remarkable that, together with the linearity, this maximum principle characterizes the diffusion operators:
Theorem. Let be the set of smooth functions and let be the set of continuous functions . Let now be an operator such that:
- is linear;
- If has a local minimum at , .
Then is a diffusion operator.
Let us consider an operator that satisfies the two above properties. As a first observation, it is readily seen from the third point that transforms constant functions into the zero function. Let now be fixed in the following argument. We are going to show that if is a smooth function, then
The idea will then be to use the Taylor expansion formula. For , the function
admits a local minimum at , thus
By letting , we therefore obtain
By considering now the function
we show in the very same way that
As a conclusion
Let now be a smooth function. By the Taylor expansion formula, there exists a smooth function such that in a neighborhood of
By applying the operator to the previous equality, and by taking account the previous observations we obtain
By denoting now,
we reach the conclusion
The matrix, is seen to be non negative, because for every ,
Now, the function is seen to attain a local minimum at , so that from the maximum principle
Finally, since transforms smooth into continuous functions, the functions ‘s and ‘s are seen to be continuous
Exercise. Let be a linear operator such that:
- is a local operator, that is if on a neighborhood of then ;
- If has a global maximum at with then .
Show that for and ,
where , and are continuous functions on such that for every , and the matrix is symmetric and nonnegative.
The previous theorem is actually a special case of a beautiful theorem that is due to Courrège that classifies the operators satisfying the positive global maximum principle. We mention this theorem without proof because the result will not be needed in the following. A complete proof may be found in the original article by Courrège.
We denote by the space of smooth and compactly supported functions . A linear operator is said to satisfy the positive maximum principle if for every function that has a global maximum at with then .
In the following statement denotes the set of Borel sets on and a kernel on is a family of Borel measures.
Theorem (Courrège’s theorem) Let be a linear operator. Then satisfies the positive maximum principle if and only if there exist functions , , and a kernel on such that for every and ,
where , with takes the constant value 1 on the ball . In addition, for every , and the matrix is a symmetric and nonnegative matrix. The functions ‘s and are continuous. Moreover for every , the function is upper semicontinuous.