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 $P^{\ast }(x,y,T)$ that the two particles are surviving at positions $x$ and $y$ at time $T$, as well as the probability ${\gamma }^{\ast }(z,t)$ 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 ${\gamma }^{\ast }(z,t)$ at position $z$ at all finite times $t\in [0,+\infty [$, whose normalization $[1-S^{\ast }(\infty )]$ determines the conditioned probability $S^{\ast }(\infty )\in [0,1]$ 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.

38 pages, 7 figures, added 3 tables. arXiv admin note: substantial text overlap with arXiv:2202.12047

References in corpus (25)

• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Dynamic first-order phase transition in kinetically constrained models of glasses
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Steady state statistics of driven diffusions
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• A minimal model of dynamical phase transition
• On the time to reach maximum for a variety of constrained Brownian motions
• Dynamical large deviations of reflected diffusions
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics
• Preface: New trends in first-passage methods and applications in the life sciences and engineering
• E. Schrödinger's 1931 paper "On the Reversal of the Laws of Nature" ["Über die Umkehrung der Naturgesetze",Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalische mathematische Klasse, 8 N9 144-153]
• First-encounter time of two diffusing particles in confinement
• Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume
• Generating discrete-time constrained random walks and Lévy flights
• Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model
• Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps
• Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment
• Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model
• Strongly constrained stochastic processes: the multi-ends Brownian bridge
• Generating stochastic trajectories with global dynamical constraints
• Generating constrained run-and-tumble trajectories
• Microcanonical conditioning of Markov processes on time-additive observables
• Inhomogeneous asymmetric exclusion processes between two reservoirs : large deviations for the local empirical observables in the Mean-Field approximation
• Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions