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?


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