A possibly Gaussian process

Question

Suppose that $X \sim N(0,1)$ and $a > 1$. Set $Y_t = X + at$. Is there a neat way to see that this defines a Gaussian process? I'd like to use it that $Y_t = 0$ with probability 1 for at least one t in the unit interval.

Answer 1

If $X>0$ then $Y_t >0$ for all $t \in [0,1]$. Hence $P\{Y_t \neq 0 \, \forall t\} \geq P\{X>0\}=\frac 1 2$. So what you are trying to prove is not correct.