**Set-up and question.**
Let $\mathcal{X}$ be a complete separable metric space which is not locally-compact. Let $V: \mathcal{X} \to [0; +\infty]$ be a function and $(X_t)_{t\geq 0}$ a Markov process in $\mathcal{X}$ such that $P\{ V(X _t) < \infty \text{ for all } t \geq 0 \} = 1$. Let $g: \mathcal{X} \to \mathbb{R}$ be such that
$$
V(X _t) - \int ^t _0 g(X _s) ds \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \quad
\quad \quad \quad \quad \quad \quad \quad
(1)
$$
is a martingale, where
$$
g(x) \leq c - V(x), \quad \text{ for all } x \in \{z \in \mathcal{X}: V(z) < \infty \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
$$
for some $c >0$.
Is it possible to deduce that
$$
\limsup\limits _{t \to \infty} \mathbb{E} V(X _t) < \infty?
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
$$

**My thoughts.** If $V$ was 'norm-like' in the sense that $\{x \in \mathcal {X} : V (x) \leq r \}$ was compact for $r >0$ and $\mathcal{X}$ was locally-compact, we would get ergodicity from (1) and (2) and thus (3) would hold. I think that without $V$ being norm-like and local compactness of $\mathcal{X}$ ergodicity cannot follow from (1) and (2), however a weaker property like (3) should hold. I have an idea how to prove (3) if additionally $(X_t)_{t\geq 0}$ is assumed to be continuous a.s. (finding an ergodic birth-death process which would stochastically dominate $(V(X_t))_{t\geq 0}$), however without continuity assumption I am currently out of ideas. I did not find any similar result in the literature, but any related reference would be kindly appreciated.

**Similar posts.** The set-up is similar to this, however the assumptions and the questions are different.

**Edit** Edited the condition on $g$. The initial version was not well formulated and resulted in a triviality as was pointed out by Iosif Pinelis.