Are $L$-diffusions unique in law? 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?
 A: Here's a common counterexample to uniqueness that usually appears in the context of SDEs, e.g. Oksendal. The equivalence of SDEs and Martingale Problems is explained in this paper by Kurtz and the more technical book Ethier & Kurtz. Put $d=1$, $a\equiv 0$, $b(x)=2|x|^{\frac{1}{2}}$ and consider the (deterministic) process $$X_t= \begin{cases}
0 & \text{if }\hspace{2mm} 0\leq t \leq \tau \\
(t-\tau)^2 & \text{if }\hspace{2mm} t>\tau
\end{cases}$$ for any $\tau>0$. Certainly $X$ is continuous and adapted and $L$ satisfies the required hypotheses. For any $f\in C^2(\mathbb{R})$ we have $f(X_t)-f(0)-\int_0^t2\sqrt{|X_s|}f'(X_s)ds=f(0)-f(0)=0$ when $t\leq\tau$. Moreover, $$f(X_t)-f(0)-\int_0^t2\sqrt{|X_s|}f'(X_s)ds=f((t-\tau)^2)-f(0)-\int_{\tau}^t2(t-\tau)f'((t-\tau)^2)ds=f((t-\tau)^2)-f(0)-\Big(f((t-\tau)^2)-f(0)\Big)=0$$ when $t>\tau$. Hence $f(X_t)-f(0)-\int_0^tLf(X_s)ds\equiv 0$ is a martingale. The fact that each $\tau>0$ leads to a different solution violates uniqueness. Note that requiring $a$ be positive definite or a Lipschitz $b$ coefficient would preclude this counterexample.
