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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.

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Take any transient, strong Feller process that satisfies your assumptions. I leave it to you to find concrete examples.

Define $\varphi(x)=\mathbb{P}_x(T<\infty)$ where $T=\inf(n\geq 0: X_n\in [0,1])$.

By transient, I mean that $\varphi(x)<1$ for some $x\in\mathbb{R}$.

Then $\varphi$ is excessive since $$\mathcal P\varphi(x)= \mathbb{P}_x(T^\prime <\infty)\leq \mathbb{P}_x(T<\infty)=\varphi(x)$$ where $T^\prime=\inf(n\geq 1: X_n\in [0,1])$.

The function $\mathcal P\varphi$ is also excessive, and continuous by the strong Feller property. Clearly $0\leq \mathcal P\varphi\leq 1$, and it is an nice exercise to show that $\mathcal P\varphi$ is not constant.

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Thanks! You're right, of course. I know examples of this ruin probability/reachability functions which have discontinuities on the boundary of a reach set. So now I just need to apply $\mathcal P$ to these functions since this operator is strongly continuous. –  Ilya Aug 25 '11 at 7:34

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