  When the unconditioned process is a diffusion submitted to a space-dependent
killing rate $k(\vec x)$, various conditioning constraints can be imposed for a
finite time horizon $T$. We first analyze the conditioned process when one
imposes both the surviving distribution at time $T$ and the
killing-distribution for the intermediate times $t \in [0,T]$. When the
conditioning constraints are less-detailed than these full distributions, we
construct the appropriate conditioned processes via the optimization of the
dynamical large deviations at Level 2.5 in the presence of the conditioning
constraints that one wishes to impose. Finally, we describe various conditioned
processes for the infinite horizon $T \to +\infty$. This general construction
is then applied to two illustrative examples in order to generate stochastic
trajectories satisfying various types of conditioning constraints : the first
example concerns the pure diffusion in dimension $d$ with the quadratic killing
rate $k(\vec x)= \gamma \vec x^2$, while the second example is the Brownian
motion with uniform drift submitted to the delta killing rate $k(x)=k
\delta(x)$ localized at the origin $x=0$.

Statistical Mechanics;Probability