Let $X(t)$ be a centered non-stationary gaussian process with covariance function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $\epsilon>0$, define
$$
p_X(\epsilon) : = \mathbb P(X(t)>0\,\forall t \in [0,\epsilon])
$$

>**Question.** *What is a good **lower-bound** for $p_X(\epsilon)$ which is valid for **small** $\epsilon$ (i.e for $\epsilon \to 0^+$) ?*

The GP I have in mind is $X(t) := tU + (1-t^2)^{1/2}V$, where $(U,V) \sim N(0,\sigma^2 I_2)$, for which the covariance function is 
$$
\rho(t,s) = ts\sigma^2 +(1-t^2)^{1/2}(1-s^2)^{1/2}\sigma^2.
$$