Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true:

$$\mathbb P\left(\forall s\in[a,b], W_s\neq 0\right)=\mathbb P\left(\sup_{ 0\leq v\leq b-a} \hat {W}_v < |W_a|\right)=\mathbb P\left(\sqrt{b-a}|Y|\leq \sqrt{a}|X|\right),$$

where $X$ and $Y$ are independent standard normal random variables. I have tried several approaches, but I think I am missing the right one.