If $f\in C_0(\mathbb R)\cap C^2(\mathbb R)$, is $bf'+\frac12\sigma^2f''\in C_0(\mathbb R)$?

Question

Let

• $b,\sigma:\mathbb R$ be Lipschitz continuous with $$|b(x)|^2+|\sigma(x)|^2\le C(1+|x|^2)\;\;\;\text{for all }x\in\mathbb R\tag1,$$ $\sigma\in C^2(\mathbb R)$, $\sigma''$ being bounded and $\sigma(\mathbb R)\subseteq\mathbb R\setminus\left\{0\right\}$
• $C_0(\mathbb R)$ denote the space of continuous functions vanishing at infinity

If $f\in C_0(\mathbb R)\cap C^2(\mathbb R)$, are we able to show that $$Lf:=bf'+\frac12\sigma^2f''\in C_0(\mathbb R)$$ or is there an example of such an $f$ with $Lf\not\in C_0(\mathbb R)$?

Answer 1

There are definitely examples of such $f$ with $Lf \not \in C_0(\mathbb{R})$. Take some $f$ that does go to $0$ at infinity but whose derivative is greater than $1$ in magnitude nearly always (one can visualize such a thing as a smoothed out version of a bunch of spikes with lengths tending to $0$). Then if we take $b \equiv 1$ and $\sigma$ some function that quickly goes to $0$ at infinity, we see $Lf \approx f'$ does not go to $0$.