Suppose $X_t$ is a continuous process in $\mathbb{R}^n$ and $f:\mathbb{R}^n\to \mathbb{R}$ is a continuous function (but not necessarily differentiable). Assuming that $M_t = f(X_t)$ is a martingale, can we apply the Ito's formula in the weak sense? In particular, I want to say that if I apply the Ito's formula, then the "$dt$" part is zero, given that we interpret the derivatives in the distributional sense.
Update: It follows from proposition 1.7 on page 262 of this book that if $A$ is the infinitesimal generator of $X_t$, then $Af = 0$. Suppose $g:\mathbb{R}^n\to \mathbb{R}$ is $C^2$ and let $\mathcal{L}$ be the differential operator satisfying $$ d g(X_t) = \mathcal{L} g(X_t) dt + (\text{"Some coefficient"})\, dW_t. $$ Then it's not hard to see that $A=\mathcal{L}$ on $C^2\cap D_A$. Here $D_A$ is the set of all the functions $g$ for which $Ag$ is well-defined. I guess now I need to show that for continuous functions $g$ in $D_A$, $Ag = \mathcal{L}g$, assuming we interpret $\mathcal{L}$ in the sense of distributions. Any thoughts?