Conditioning diffusion processes with killing rates

When the unconditioned process is a diffusion submitted to a space-dependent killing rate k((x) over right arrow), 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 is an element of [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 -> +infinity. 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((x) over right arrow)=gamma(x) over right arrow (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.

Automated summary

This study focuses on imposing various conditioning constraints on a diffusion process with a space-dependent killing rate for a finite or infinite time horizon. The conditioned processes are constructed through optimization of dynamical large deviations under the desired conditioning constraints. Illustrative examples are provided to generate stochastic trajectories that satisfy different types of conditioning constraints.