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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.

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