Given two Markov chains $\left(X^{(1)}_n\right)_{n\in\mathbb N_0}$ and $\left(X^{(2)}_n\right)_{n\in\mathbb N_0}$ with transition kernel $\kappa_1$ and $\kappa_2$, respectively, and a common stationary distribution $\pi$, we say that $\left(X^{(1)}_n\right)_{n\in\mathbb N_0}$ converges faster than $\left(X^{(2)}_n\right)_{n\in\mathbb N_0}$ if $$\sup_A\left|\kappa_1^n(x,A)-\pi(A)\right|\le\sup_A\left|\kappa_2^n(x,A)-\pi(A)\right|\tag1$$ for all $x$ and $n$.
How does this notion carry over to continuous-time Markov processes?
To be precise: Assume $(U_t^c)_{t\ge0}$ is a solution of $${\rm d}U_t^c=\frac c2h'(U_t^c){\rm d}t+\sqrt c{\rm d}W_t\tag2$$ with $c>0$, $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and a Brownian motion $W$.
Assume $h=\ln f$ for some positive $f\in C^2(\mathbb R)$. It's known that $U^c$ has stationary measure $\mu:=f\lambda^1$ (measure with density $f$ wrt the Lebesgue measure $\lambda^1$; assume $\int f\:{\rm d}\lambda^1=1$).
I've read that in the sense of the notion above, $U^c$ converges faster than $U^{c'}$ if $c>c'$. How has this to be understood?
Clearly, $(1)$ is the total variation distance of the $n$th power of the transition kernel and the stationary distribution $\pi$. So, I suppose it's meant that we now consider $\sup_A\left|\kappa_t^{(c)}(x,A)-\mu(A)\right|$, where $\left(\kappa^{(c)}_t\right)_{t\ge0}$ is the transition semigroup of $U^{(c)}$. If that's what is meant, how do we see that this quantity (for fixed $x$ and $t$) becomes larges as $c$ becomes larger?
This is not an answer to your exact question and I am ignoring quite some details, but maybe it helps anyway. The generator of your process can be written as $$ \mathcal L_c f=\frac{c}{2}(f'' +h'f'), \quad f\in C_c^2(\Bbb R),$$ and with $\mu=e^h\lambda$ you can check that $$ \mathcal E_c(f):=-\int f \mathcal L_c fd\mu=\frac{c}{2}\int (f')^2d\mu.$$ If $h$ is concave and $\int fd\mu=0$, a classical result by Bobkov yields $$ \int f^2d\mu \le C_1 \int (f')^2d\mu$$ for some $C_1>0$. Together, we have the Poincaré inequality $$ \|f\|^2_{L^2(\mu)}\le\frac{2C_1}{c}\mathcal E_c(f) ,$$ which implies (by taking derivatives and using Gronwall's Lemma) $$ \|\kappa^{(c)}_t(\,\cdot\,, f)\|_{L^2(\mu)}\le e^{- C_2 ct} \|f\|_{L^2(\mu)}.$$ This $L^2$-rate for the decay of the semi-group is indeed faster, if one increases $c$.