I've been trying to understand diffusions. We can show they exist by noting they solve particular SDEs, but are they unique? More precisely:
Fix a filtered probability space satisfying the usual conditions and locally bounded measurable functions $a_{i,j}$ and $b$ from $\mathbb{R}^d$ to $\mathbb{R}$, such that $(a_{i,j}(x))_{i,j}$ is symmetric non-negative definite for all $x\in\mathbb{R}^d$. Define the operator $L$ by
$Lf(x)=\frac{1}{2}\sum_{i,j}a_{i,j}(x)\frac{\partial f}{\partial x_ix_j}+\sum_ib_i(x)\frac{\partial f}{\partial x_i}$
Suppose that the continuous, adapted process $X$ is such that, for all $f\in C^2(\mathbb{R}^d)$,
$f(X_t)-f(0)-\int_0^tLf(X_s)ds$
is a local martingale, and $X_0=0$.
Then is $X$ unique in law?