TITLE: Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz
QUESTION [0 upvotes]: Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'\psi)'\;\;\;\text{for }\psi'\in C^2(\mathbb R).$$ Note that $L^\ast e^{-h}=0$.

Are we able to show that there is a Borel measurable (hopefully continuous) function $v:\mathbb R\to[0,\infty)$ such that $$Lv\le c-\lambda v\tag1$$ for some $c\ge0$ and $\lambda>0$?

REPLY [1 votes]: Suppose that $h'(x)$ satisfies a dissipativity condition: there exists $K>0$ and $A \ge 0$ such that for all $x \in \mathbb{R}$ we have $x h'(x) \ge K x^2 - A$.  Consider as a candidate function $v(x) = x^2$.  Then, $$
L v(x) = - x h'(x) + 1 \le - K x^2 + A + 1 \;.
$$ So, (1) is satisfied with $\lambda=K$ and $c=A+1$.  
Other conditions are possible.  See, e.g., Assumption 2.1 (B) and Lemma 2.5 of Non-Asymptotic Mixing of the MALA Algorithm.
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