Theorem 3.4.3.label Let $a: \real^{n} \to L(\real^{n}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be bounded measurable functions such that the martingale problem for $(a, b)$ is well-posed, that is, let
\[Lu = \frac{1}{2}\dpn{A, D^2f}{\real^{n \times n}}+ \dpn{b, Df}{\real^n}\]
then for each $y \in \real^{n}$, there exists a unique probability measure $\bp^{y}$ on $C([0, \infty); \real^{n})$ such that:
- (1)
$\bp^{y}(x_{0} = y) = 1$.
- (2)
For each $f \in C_{c}^{\infty}$,
\[C_{t}^{f} = f(\pi_{t}) - f(x_{0}) - \int_{0}^{t} Lf(x_{s})ds\]is a $\bp^{y}$-martingale.
then
- (1)
$\bracs{x_t|t \ge 0}$ is a time-homogeneous strong Markov process with respect to $\bp^{y}$.
- (2)
If $a = \sigma \sigma^{*}$ and $(\sigma, b)$ satisfy the Lipschitz condition, then the generator of the above process is $L$.