Let $(X(t))_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t_0 \in (-1,1)$ and $\epsilon \in (0,1-t_0)$, define $$ p_X(t_0,\epsilon) : = \mathbb P(X(t)>0\,\forall t \in [t_0,t_0+\epsilon]) $$

Question. What is a good lower-bound for $p_X(t_0,\epsilon)$ which is valid for small $\epsilon$ (i.e for $\epsilon \to 0^+$) ?

A concrete example. 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 covariation function is $$ \rho(t,s) = ts\sigma^2 +(1-t^2)^{1/2}(1-s^2)^{1/2}\sigma^2. $$ In case it helps, $\sigma^2$ can be taken to be very small.