Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE

Let

• $$h\in C^2(\mathbb R)$$ with $$h''\ge\rho\tag1$$ for some $$\rho>0$$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
• $$\mu$$ be the measure with density $$\varrho$$ with respect to the Lebesgue measure $$\lambda$$ on $$(\mathbb R,\mathcal B(\mathbb R))$$
• $$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$$ be a continuous process on a probability space $$(\Omega,\mathcal A,\operatorname P)$$ with $$X^x_t=x+\int_0^t\frac12(\ln\varrho)'(X^x_s)\:{\rm d}s+W_t\;\;\;\text{for all }t\ge0\text{ almost surely for all }x\in\mathbb R\tag2$$ 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 $$\Gamma(f):=\frac12|f'|^2\;\;\;\text{for }f\in C^1(\mathbb R)$$ and $$Lf:=-\frac12h'f'+\frac12f''\;\;\;\text{for }f\in C^2(\mathbb R).$$

How can we show that $$\Gamma(\kappa_tf)\le e^{-\rho t}\kappa_t(\Gamma(f))\tag3$$ for all $$f$$ belonging to a suitable class $$\mathcal C$$ of functions $$f:\mathbb R\to\mathbb R$$ (I hope for $$\mathcal C=C_c^\infty(\mathbb R)$$) for all $$t\ge0$$?

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 $$\left.A\right|_{\tilde{\mathcal D}(A)}=\left.L\right|_{\tilde{\mathcal D}(A)}\tag4.$$ Moreover, $$\mu$$ is reversible (and hence invariant) with respect to $$(\kappa_t)_{t\ge0}$$. By $$(1)$$, $$\Gamma_2(f):=\frac14(|f''|^2+h''|f'|^2)\ge\frac\rho2\Gamma(f)\;\;\;\text{for all }f\in C_c^\infty(\mathbb R)\tag5.$$

In order for $$(3)$$ to make sense, we need to prove that $$\kappa_tf$$ belongs to the domain of $$\Gamma$$ (otherwise, the left-hand side would be undefined) for all $$f\in\mathcal C$$ and $$t\ge0$$.

We know that $$(\kappa_tf)(x)=f(x)+\int_0^t\left(\kappa_s\left(Lf\right)\right)(x)\:{\rm d}s\;\;\;\text{for all }(t,x)\in[0,\infty)\times\mathbb R\tag6$$ for all $$f\in C_b^2(\mathbb R)$$.