Answer by RaphaelB4 for Strong Markov property, independence, regular conditional probability

Strong Markov property says that under the condition that $\forall i, y(\tau_i)=y_i$, on each $[\tau_i,\tau_{i-1}]$ the process $X_t$ is just a brownian motion starting at $(i,y_i)$ and ending at $(i-1,y_{i-1})$. This evolution only depend on the starting point and the ending point and not at all what append before $\tau_i$ or after $\tau_{i-1}$. Then $X_t$ on $[\tau_i,\tau_{i-1}]$ and $[\tau_j,\tau_{j-1}]$ are independant (for $j\neq i$).