# How is integration by parts applied here?

Let

• $$C_0(\mathbb R)$$ denote the space of continuous functions vanishing at infinity equipped with the supremum norm
• $$b,\sigma:\mathbb R\to\mathbb R$$ be Lipschitz continuous, $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ and $$L^\ast g:=\frac12(\sigma^2g)''-(bg)'\;\;\;\text{for }g\in C^2(\mathbb R)$$
• $$A$$ be a closed linear operator on $$C_0(\mathbb R)$$ with $$A\phi=L\phi\;\;\;\text{for all }\phi\in C_c^2(\mathbb R)\tag1$$
• $$\lambda^1$$ denote the Lebesgue measure
• $$g\in C^2(\mathbb R)$$ with $$g\ge0$$, $$\int g\;{\rm d}\lambda^1=1$$ and $$\int\phi L^\ast g\:{\rm d}\lambda^1=0\;\;\;\text{for all }\phi\in C_c^\infty(\mathbb R)\tag2$$
• $$\mu$$ denote the measure with density $$g$$ with respect to $$\lambda^1$$

I want to show that $$\int Af\:{\rm d}\mu=0\tag3\;\;\;\text{for all }f\in C_0(\mathbb R).$$

By $$(1)$$ and $$(2)$$, we should obtain $$\int A\phi\:{\rm d}\mu=\int (L\phi)g\:{\rm d}\lambda^1\stackrel{\text{(4)}}=\int\phi(L^\ast g)\:{\rm d}\lambda^1\tag5=0\;\;\;\text{for all }\phi\in C_c^\infty(\mathbb R)$$ (Could anyone point me to a reference for $$(4)$$?). Now, let $$f\in C_0(\mathbb R)$$. Since, $$C_c^\infty(\mathbb R)$$ is dense in $$C_0(\mathbb R)$$ (again, any reference for that?), there is a $$(\phi_n)_{n\in\mathbb N}\subseteq C_c^\infty(\mathbb R)$$ with $$\left\|\phi_n-f\right\|_\infty\xrightarrow{n\to\infty}0\tag6.$$

How can we conclude?

Remark: Any idea for a better title? The problem is occuring in the infinitesimal characterization of invariant distributions for diffusion processes on $$\mathbb R$$.

• Reference for density: Corollary 1, page 159, in Treves book. – Pedro Dec 26 '18 at 17:52
• I could be wrong here but I believe you might be missing the hypothesis: $\mathcal{C}_c^\infty(\mathbb{R})$ is a core for the operator $A$ which would allow you to conclude. – Michh Dec 28 '18 at 3:16