  We consider two independent identical diffusion processes that annihilate
upon meeting in order to study their conditioning with respect to their
first-encounter properties. For the case of finite horizon $T<+\infty$, the
maximum conditioning consists in imposing the probability $P^*(x,y,T ) $ that
the two particles are surviving at positions $x$ and $y$ at time $T$, as well
as the probability $\gamma^*(z,t) $ of annihilation at position $z$ at the
intermediate times $t \in [0,T]$. The adaptation to various conditioning
constraints that are less-detailed than these full distributions is analyzed
via the optimization of the appropriate relative entropy with respect to the
unconditioned processes. For the case of infinite horizon $T =+\infty$, the
maximum conditioning consists in imposing the first-encounter probability
$\gamma^*(z,t) $ at position $z$ at all finite times $t \in [0,+\infty[$, whose
normalization $[1- S^*(\infty )]$ determines the conditioned probability
$S^*(\infty ) \in [0,1]$ of forever-survival. This general framework is then
applied to the explicit cases where the unconditioned processes are
respectively two Brownian motions, two Ornstein-Uhlenbeck processes, or two
tanh-drift processes, in order to generate stochastic trajectories satisfying
various types of conditioning constraints. Finally, the link with the
stochastic control theory is described via the optimization of the dynamical
large deviations at Level 2.5 in the presence of the conditioning constraints
that one wishes to impose.
