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 $$ \rho(t,s) = ts\sigma^2 +(1-t^2)^{1/2}(1-s^2)^{1/2}\sigma^2. $$