# Stability of the Langevin semigroup under $C_c^\infty(\mathbb R)$

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.