I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be computed easily. For instance on the figure, the particle in dark blue hits at $t_{1}$ the boundary $B_{1}$ and dies. But my problem has multiple boundaries (as in the figure) and I wanted to know if there was a clever way to compute the probability for a particle to hit the second boundary $B_{2}$ given that it avoided $B_{1}$. I found a reference on approximation of boundary crossing problems but it is not exactly what I am looking for. There is also the running maximum of a Brownian motion but it is taken over a period of time. My guess is that at $B_{1}$ we have a normal distribution which will be truncated at $B_{2}$, given that some particles have hit $B_{1}$. And another truncation at $B_{3}$ on the same idea, etc... .
Blue lines are Brownian motions, boundaries do not need to have the same height.
To summarize my question: can we compute (analytically/closed-form) the probability that a particle hits only, for example, the second boundary $B_{2}$ at time $t_{2}$ and has not touched $B_{1}$ at $t_{1}$? Can it be generalized to the case that the particle hits only $B_{n}$?
Thanks in advance for your answers.