= Solution
For the diffusion operator
$$
Lf=\tfrac12\sum_{i,j}a_{ij}\partial_{ij}f+\sum_i b_i\partial_i f,
$$
a continuous adapted process solves the <diffusion martingale problem> when, for every $f\in C_b^2(\mathbb R^d)$,
$$
\boxed{f(X_t)-f(X_0)-\int_0^tLf(X_s)\,ds\text{ is a true martingale}.}
$$
Here $C_b^2$ means that the function and its derivatives through order two are bounded and continuous. An equivalent compact-support/local convention is often used for the general <martingale problem>; the bounded-coefficient setting lets us use the true-martingale convention explicitly. The matrix $a$ is interpreted as a covariance matrix in the diffusion formulation, hence is symmetric nonnegative; the displayed operator in any case uses only its symmetric part.
A <well-posed martingale problem> has existence and uniqueness in law on continuous path space for each specified starting point, or for each initial law in the initial-law formulation. This is uniqueness of the process law, not an assertion of pathwise uniqueness on a prescribed Brownian probability space.
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