Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
No, hitting times for the same Brownian Motion are not independent. Think of it this way: given $T_a = s$, $B_s = a$ so if $a \ne b$, $B_t$ is much less likely to be close to $b$ when $t$ is near $s$, and therefore $T_b$ is much less likely to be close to $s$.