I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ):


let X being a stochastic process with almost surely continuous sample path, and such that, there exists a constant $\eta>0$ such that,


$
\forall \epsilon>0,\ \exists \mathcal{C}\_{\epsilon}>0, \ \mathbb{P}(|X\_t|>\epsilon)\leq \mathcal{C}\_{\epsilon} e^{-\eta t}.
$


Then, $X\_t$ converge almost surely to zero!


In discrete time, such question is easly solved using Borel-Cantelli Lemma. How to make it works in continuous time? Else, have you a counter-exemple? In all my trials, I wasn't be be able to prove this statement without using some kind of uniform continuity...


Thank you!



Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process?
Consider a sequence of independent uniform(0,1) random variables $U\_j$, and
let $$X\_t = \sum\_{j=1}^\infty \max(0,1 - 2^j |t - j - U\_j|)$$ In particular
$X\_t = 0$ outside the intervals $(j + U\_j - 2^{-j}, j + U\_j + 2^{-j})$, and so
$P(X\_t > 0) < 2^{2-t}$. But of course $X\_t$ does not converge to $0$. 


