Consider general Brownian bridge W(0)=0; W(T) = a. (Here "general" means: $W(T)\ne 0$).

What is the probability W(t) >= b, for all $ t \in [0, T] $ ?

Is there close simple formula in terms of a, b , T ?

Can the formula be the same for any martingale type stochastic process (e.g. random walk), not only brownian motion, or it somehow depends on the details of stochastic process ? If there is such dependence how the questioned probability will change if we consider the distribition W(t) to have more and more "heavy tails" ?

(I'm sure that questions are well-known for experts, but it is somehow difficult to google the asnwer, so let me ask here).

Remark: If we consider somewhat informally related question - probability for W first hit a, before hitting (-b), (i.e. W(some T)=a and W(t < T) > b) there is simple formula P = b/(a+b), which holds true for any martingale stochastic process. The questions are somewhat different, but still resemble each other and so the simplicity of the answer P=b/(a+b) makes me hope that the answer to my question might be simple and closed form.