Distribution of minimum of independent normal variables

Question

Suppose $X_t\sim N(\mu,\sigma^2t)$, $X_t$ are independent. Is the distribution of $$\min_{0\leq t\leq T}X_t$$known? In other words can this probability be found $$P(\min_{0\leq t\leq T}X_t\leq a)?$$ I have seen some results about the $\min(X,Y)$ but this is not exactly what I am looking for.

Answer 1

$$P(\min_{0\leq t\leq T}X_t\leq a) =1 - P(all X_t >a)$$

Since $X_t$ are all independent, then this equals:

$$1 - P(X_1>a)*P(X_2 >a)*...*P(X_T>a)$$

You can use that to find the pdf.