I am aware that this question is rather unspecific, but I was curious what versions of the Ito formula for Sobolev functions exist (and what methods are used to prove them).The only result I am aware of is the following, but I would be interested in versions with (maybe) even more relaxed assumptions.
Let $f\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$ with $p>1+\frac{n}{2}$, then we have $$f(W_{t})=f(0)+\int_{0}^{t}\nabla f(W_{s})\cdot dW_{s}+\frac{1}{2}\int_{0}^{t}\varDelta f(W_{s})\:dW_{s},$$ where $W_{\cdot}$ is an $n$-dimensional Brownian motion.
Proof:We define $\rho(t,x)=\frac{1}{(2\pi t)^{\frac{n}{2}}}\exp\left(-\frac{\vert x\vert^{2}}{2t}\right)$. Let us first consider the case $n>2$. Let $f_{n}$ be a regularizing sequence for $f$, which we obtained by a convolution with the usual mollifier. Then $f_{n}\in C^{\infty}(\mathbb{R}^{n})$ and $f_{n}$ converges to $f$, uniformly on compact sets, so that $$\lim_{n\rightarrow\infty}f_{n}(W_{t})=f(W_{t}),$$ for any $t\geq0$. By the standard Ito formula, we have $$f_{n}(W_{t})=f_{n}(W_{0})+\int_{0}^{t}\nabla f_{n}(W_{s})\cdot dW_{s}+\frac{1}{2}\int_{0}^{t}\varDelta f_{n}(W_{s})\:ds.$$ Further, by the Ito isometry, $$\mathbb{E}\left[\left(\int_{0}^{t}(\nabla f_{n}(W_{s})-\nabla f(W_{s}))\cdot dW_{s}\right)^{2}\right]$$ $$=\int_{0}^{t}\mathbb{E}\left[\vert\nabla f_{n}(W_{s})-\nabla f(W_{s})\vert^{2}\right]\:ds$$ $$=\int_{0}^{t}\int_{\mathbb{R}^{n}}\vert\nabla f_{n}(x)-\nabla f(x)\vert^{2}\rho(s,x)\:dx\:ds=:I_{n}.$$
If $p>n$, we have $\lim_{n\rightarrow\infty}I_{n}=0$ by Lebesgue's theorem, since $\nabla f\in C\cap L^{\infty}$, and so the integrand converges to zero pointwise and is dominated by the integrable function $\|\nabla f_{n}-\nabla f\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\rho$ . On the other hand, if $1+\frac{n}{2}<p\leq n$, we have $\vert\nabla f\vert^{2}\in L^{q}(\mathbb{R}^{n})$ for some $q>1+\frac{n}{2}$. Let $q'$ be the conjugate exponent of $q$, then we have $$q'=1+\frac{1}{p-1}<1+\frac{2}{n}$$ and therefore, as $\rho\in L^{p}$ for any $p\in(0,1+\frac{2}{n})$, $\rho\in L^{q'}((0,T)\times\mathbb{R}^{n})$. By Hölder's inequality, we obtain
$$I_{n}\leq\left\Vert \vert\nabla f_{n}-\nabla f\vert^{2}\right\Vert _{L^{q}((0,T)\times\mathbb{R}^{n})}\|\rho\|_{L^{q'}((0,T)\times\mathbb{R}^{n})}\rightarrow_{n\rightarrow\infty}0.$$ Finally, $$\mathbb{E}\left[\left\vert \int_{0}^{t}(\varDelta f_{n}(W_{s})-\varDelta f(W_{s}))\:ds\right\vert \right]$$ $$\leq\int_{0}^{t}\mathbb{E}\left[\left\vert \varDelta f_{n}(W_{s})-\varDelta f(W_{s}))\right\vert \right]\:ds$$ $$=\int_{0}^{t}\int_{\mathbb{R}^{n}}\vert\varDelta f_{n}(x)-\varDelta f(x)\vert\rho(s,x)\:dx\:ds.$$ By applying Hölder's inequality with the conjugate exponent of $p$, which we call $p'$, the last term is
$$\leq\|\varDelta f_{n}-\varDelta f\|_{L^{p}((0,T)\times\mathbb{R}^{n})}\|\rho\|_{L^{p'}((0,T)\times\mathbb{R}^{n})}\rightarrow_{n\rightarrow\infty}0,$$ since $f_{n}$ converges to $f$ in $W^{2,p}(\mathbb{R}^{n})$ and the assumption $p>1+\frac{2}{n}$ implies $p'<1+\frac{2}{n}$. We have, as as $\rho\in L^{p}$ for any $p\in(0,1+\frac{2}{n})$, $\|\rho\|_{L^{p'}((0,T)\times\mathbb{R}^{n})}<\infty.$
In conclusion, we showed that the proposed form of Ito's formula holds a.s. for every t>0. In the case $n\leq2$, the hypothesis $p>1+\frac{n}{2}$ implies that $p>n$ and the claim can be proved as before.
