Let

- $h\in C^2(\mathbb R)$
- $(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$X^x_t=x-\frac12\int_0^th'(X^x_s)\:{\rm d}s+W_t\;\;\;\text{for all }t\ge0\text{ almost surely for all }x\in\mathbb R\tag1$$ for some Brownian motion $(W_t)_{t\ge0}$ on $(\Omega,\mathcal A,\operatorname P)$,$$\kappa_t(x,B):=\operatorname P\left[X^x_t\in B\right]\;\;\;\text{for }(x,B)\in\mathbb R\times\mathcal B(\mathbb R)$$ and $$\kappa_tf:=\int\kappa_t(\;\cdot\;,{\rm d}y)f(y)$$ for Borel measurable $f:\mathbb R\to\mathbb R$ with $\kappa_t|f|<\infty$ for $t\ge0$

Now, let $$Lf:=-\frac12h'f'+\frac12f''\;\;\;\text{for }f\in C^2(\mathbb R).$$

We know that $(\kappa_t)_{t\ge0}$ is a strongly continuous contraction semigroup on $C_0(\mathbb R)$. If $(\mathcal D(A),A)$ is the corresponding generator, then $$\tilde{\mathcal D}(A):=\left\{f\in C_0(\mathbb R)\cap C_b^2(\mathbb R):Lf\in C_0(\mathbb R)\right\}\subseteq\mathcal D(A)$$ and $$\left.A\right|_{\tilde{\mathcal D}(A)}=\left.L\right|_{\tilde{\mathcal D}(A)}\tag2.$$

Moreover, $$(\kappa_tf)(x)=f(x)+\int_0^t\left(\kappa_s\left(Lf\right)\right)(x)\:{\rm d}s\;\;\;\text{for all }(t,x)\in[0,\infty)\times\mathbb R\tag3$$ for all $f\in C_b^2(\mathbb R)$.

If $f\in C_c^k(\mathbb R)$ for some $k\in[1,\infty]$, are we able to show that $\kappa_tf\in C^l(\mathbb R)$ for some $l\in[1,\infty]$?

I assume that stability of $(\kappa_t)_{t\ge0}$ under any of the classes $C_c^k(\mathbb R)$ depends on the smoothness of the coefficients of $L$ (in this case, $h$), but I don't know how we need to argue.