Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have $ \mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2 $. Here $\kappa > 0$ is a constant.

Question: Conditioning on that the realization of $Z_0$ is sufficiently small, can we prove that $\mathbb{E}\{Z_{t}\} \leq c\exp(-t^2)$, where $c$ is an constant, or something like $\mathbb{E}\{Z_{t}^2\} \leq c\exp(-t^2)$? If not, what additional conditions on $\kappa$ or the value of $Z_0$ do we need? Or is there any counter example for this claim?

If we can further assume the boundedness of $Z_t$ for all $t$, (how) can we prove this claim?