Conditioning diffusion processes with killing rates

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