# Function of transient markov processes

Suppose that $$X_t$$ and $$Z_t$$ are Markov processes, which are $$\mathbb{P}\otimes m$$-a.e. (here $$m$$ is the Lebesgue measure on $$\mathbb{R}$$) not equal. If $$X_t$$ and $$Z_t$$ are transient, it seems right that $$\mathbb{P}\left(\|X_t-Z_t\|^2>x\right)>0;\qquad (\forall x \geq 0).$$

How do I prove this, since any large deviation result I know proves an aupper bound on this inequality?