  When the unconditioned process is a diffusion living on the half-line $x \in
]-\infty,a[$ in the presence of an absorbing boundary condition at position
$x=a$, we construct various conditioned processes corresponding to finite or
infinite horizon. When the time horizon is finite $T<+\infty$, the conditioning
consists in imposing the probability $P^*(y,T ) $ to be surviving at time $T$
and at the position $y \in ]-\infty,a[$, as well as the probability
$\gamma^*(T_a ) $ to have been absorbed at the previous time $T_a \in [0,T]$.
When the time horizon is infinite $T=+\infty$, the conditioning consists in
imposing the probability $\gamma^*(T_a ) $ to have been absorbed at the time
$T_a \in [0,+\infty[$, whose normalization $[1- S^*(\infty )]$ determines the
conditioned probability $S^*(\infty ) \in [0,1]$ of forever-survival. This case
of infinite horizon $T=+\infty$ can be thus reformulated as the conditioning of
diffusion processes with respect to their first-passage-time properties at
position $a$. This general framework is applied to the explicit case where the
unconditioned process is the Brownian motion with uniform drift $\mu$ in order
to generate stochastic trajectories satisfying various types of conditioning
constraints. Finally, we describe the links with the dynamical large deviations
at Level 2.5 and the stochastic control theory.
