Conditioning diffusion processes with killing rates

TL;DR

This paper develops a framework for constructing conditioned diffusion processes with space-dependent killing rates under various constraints, including finite and infinite horizons, using large deviations principles.

Contribution

It introduces a general method to build conditioned diffusions with killing, extending previous models to more complex and realistic scenarios with space-dependent killing rates.

Findings

Constructed conditioned processes for finite and infinite horizons.

Applied the framework to diffusion with quadratic killing rate.

Analyzed Brownian motion with delta killing at the origin.

Abstract

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 $x=0$.