In my research, all the proof comes down to an estimate of the following term


$$\int\_t^{t+h} E|\partial\_x P\_{t+h-\tau}f(X\_\tau)-P\_{t+h-\tau}\partial\_xf(X\_\tau)|^2\,d\tau,\tag{1}$$
where $t>0$ is fixed, $P\_t$ is the transition semigroup associated with a time-homogeneous Ito diffusion, i.e., SDE with time-independent Lipschitz coefficients and $\partial\_x P\_{t+h-\tau}f(X\_\tau)$ is interpreted as plugging $X\_\tau$ into the function $(\partial\_x P\_{t+h-\tau}f)(\cdot)$.


By assuming $f$ is a fixed regular enough function, I hope to get an estimate of (1) in terms of $h$. In my opinion, the problem is how to estimate 
$$|\partial\_x P\_{t+h-\tau}f-P\_{t+h-\tau}\partial\_xf|\tag{2}.$$


Suppose $X\_t$ has a symmetric transition density (for instance, the associated elliptic operator $L$ of $X\_t$ is of divergence form), then using the definition of transition semigroup, we can show (2) is always zero, hence (1) is zero.


On the other hand, suppose $L$ is not of divergence form, intuitively, if $f$ is good enough, by strong continuity of semigroup, $P\_{t+h-\tau}f\to f$ as $h\to 0$ which could gives us
$$\partial\_x P\_{t+h-\tau}f\to \partial\_x f$$
thus $(1)\to 0$ intuitively. 


Are there any works about this topic? Can I possibly get a convergence rate for (1)?



relationship between transition semigroup and first order spatial derivative
It seems one can obtain the required estimate by a straightforward application of [Dynkin's formula](https://en.wikipedia.org/wiki/Dynkin%27s_formula). In particular, Dynkin's formula allows you to expand $\partial\_x P\_h f(x)$ about $h=0$, $$
\partial\_x P\_h f(x) = \partial\_x f(x) + \int\_0^h \partial\_x \mathbb{E}\_x L f(X\_s) ds
$$ where $L$ is the infinitesimal generator of $X$. You can iterate this formula to obtain a better rate of convergence, meaning reapply it to $Lf(x)$ in place of $f(x)$.


