Consider a discrete time Feller Markov process $X$ on $\mathbb R$ with a kernel $K(x,dy) = \xi(x,y)dy$ and the transition operator $ \mathcal Pf(x) = \int\limits_{\mathbb R}f(y)\xi(x,y)\,dy. $
Here $\xi$ is a continuous and strictly positive function.
A function $f$ is excessive if $\mathcal Pf\leq f$, or equivalently if the process $f(X)$ is a supermartingale. I wonder if there are non-constant excessive functions which are bounded and continuous.