Diffusion generators and martingale problems

For a diffusion with drift vector $b$ and noise matrix $\sigma$, the covariance matrix is $a=\sigma\sigma^{\mathsf T}$. Apply Itô's formula to a twice continuously differentiable test function. Its finite-variation term is the gradient paired with the drift plus one half of the Hessian contracted with covariance. This defines the local differential expression $L$. The number and coordinates of Brownian drivers do not enter separately once their resulting covariance has been formed.

Subtracting the integral of $Lf$ from the test function along the process leaves a local martingale. For compactly supported twice differentiable tests and locally bounded coefficients on the whole Euclidean space, the Brownian integrand is bounded, yielding a true martingale on finite horizons. Unbounded tests, such as a polynomial, require stopping and moment estimates before expectations are taken. The formal expression can be correct even when an unjustified expectation calculation based on it is not.

If the transition operators form a strongly continuous Feller semigroup on the continuous functions vanishing at infinity, its strong generator is defined by a uniform-norm limit at time zero. The domain consists of the functions for which that limit exists in the same function space. A smooth differential expression alone is not the whole operator. Whether a particular test belongs to the strong-generator domain depends on its behavior at infinity, regularity, and any boundary conditions.

The same interior expression on a half-line can accompany different processes. Killed Brownian motion at zero uses an absorbing Dirichlet condition on the interior state space, while reflecting Brownian motion has an appropriate Neumann derivative condition at the boundary. Reflection also introduces a boundary local-time term in a trajectory description. These are not interchangeable solutions of an unconstrained whole-line SDE simply because the second derivative in the interior is identical.

The martingale problem specifies an initial distribution and asks for a probability law on continuous-path space such that $M^f$ is a martingale for every test in a declared class, commonly smooth compactly supported functions. Existence and uniqueness of this law mean the martingale problem is well posed. An SDE solution supplies such a law through Itô's formula. Under suitable coefficient and covariance-factorization assumptions, the converse can be realized on an enlarged probability space, which is a weak rather than automatically strong construction.

For globally Lipschitz diffusion coefficients, the previous existence and uniqueness theory supplies a well-posed corresponding diffusion law. For rough coefficients, the martingale problem becomes a useful independent formulation, but well-posedness must still be proved. Merely writing $Lf$ does not guarantee a process exists or that its law is unique. The framework separates the local differential data from the probabilistic consistency and uniqueness questions that those data must satisfy.