Conditioning two diffusion processes with respect to their first-encounter properties

Abstract

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.

Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes

“…Over the years, many different conditioning constraints have been considered, including the Brownian excursion [26,27], the Brownian meander [28], the taboo processes [29][30][31][32][33][34], or nonintersecting Brownian bridges [35]. Let us also mention the conditioning in the presence of killing rates [18,[36][37][38][39][40][41][42][43] or when the killing occurs only via an absorbing boundary condition [44][45][46][47]. Stochastic bridges have been also studied for many other Markov processes, including various diffusions processes [48][49][50], discrete-time random walks and Lévy flights [51][52][53], continuous-time Markov jump processes [53], run-and-tumble trajectories [54], or processes with resetting [55].…”

Conditioning diffusion processes with killing rates

“…The goal of the present paper is thus to revisit the conditioning of diffusion processes with space-dependent killing rates, in order to give a global discussion of the various conditioning constraints that can be imposed for finite horizon T or for infinite horizon T = +∞. It will be also interesting to see the similarities and the differences with the cases where the diffusion process is killed only via an absorbing boundary condition [49][50][51][52].…”