# Show that $C_c^∞(\mathbb R)$ is a core of the generator of the Feller semigroup induced by the strong solution of an SDE with Lipschitz coefficients

Let

• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$b,\sigma:\mathbb R\to\mathbb R$$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$
• $$W$$ be a Brownian motion on $$(\Omega,\mathcal A,\operatorname P)$$
• $$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$$ be a continuous process on $$(\Omega,\mathcal A,\operatorname P)$$ with $$X_t^x=x+\int_0^tb(X^x_s)\:{\rm d}s+\int_0^t\sigma(X^x_s)\:{\rm d}W_s\;\;\;\text{for all }t\ge0\text{ almost surely for all }x\in\mathbb R\tag1$$ and $$(\kappa_tf)(x):=\operatorname E\left[f(X^x_t)\right]\;\;\;\text{for }x\in\mathbb R\tag2$$ for any bounded Borel measurable $$f:\mathbb R\to\mathbb R$$ and $$t\ge0$$

Via $$(2)$$, $$(\kappa_t)_{t\ge0}$$ is a strongly continuous contraction semigroup on $$C_0(\mathbb R)$$. Let $$(\mathcal D(A),A)$$ denote the generator of that semigroup. As discussed in the comments of the answer to my other question, $$\left\{f\in C_0(\mathbb R)\cap C_b^2(\mathbb R):Lf\in C_0(\mathbb R)\right\}\subseteq\mathcal D(A).\tag3$$

Question 1: How can we show that $$C_c^\infty(\mathbb R)$$ is a core of $$(\mathcal D(A),A)$$.

Question 2: How can we show that $$\tilde{\mathcal D}(A):=\left\{f\in C_0(\mathbb R)\cap C^2(\mathbb R):Lf\in C_0(\mathbb R)\right\}=\mathcal D(A)\tag4,$$ or at least that the left-hand side is contained in $$\mathcal D(A)$$?

The result I asked for in the second question seems to be (i.e. I'm not sure) a consequence of Theorem 2.1 of Chapter 8 in Markov Processes: Characterization and Convergence, at least under the assumption that $$\sigma(\mathbb R)\subseteq\mathbb R\setminus\left\{0\right\}$$ and $$\sigma\in C^2(\mathbb R)$$ such that $$\sigma''$$ is bounded.