Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Abstract

When the unconditioned process is a diffusion living on the half-line $x\in ]-\infty ,a[$ in the presence of an absorbing boundary condition at position $x=a$, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite $T<+\infty$, the conditioning consists in imposing the probability $P^{\ast }(y,T)$ to be surviving at time $T$ and at the position $y\in ]-\infty ,a[$, as well as the probability ${\gamma }^{\ast }(T_a)$ to have been absorbed at the previous time $T_a\in [0,T]$. When the time horizon is infinite $T=+\infty$, the conditioning consists in imposing the probability ${\gamma }^{\ast }(T_a)$ to have been absorbed at the time $T_a\in [0,+\infty [$, whose normalization $[1-S^{\ast }(\infty )]$ determines the conditioned probability $S^{\ast }(\infty )\in [0,1]$ of forever-survival. This case of infinite horizon $T=+\infty$ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passage-time properties at position $a$. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift $\mu$ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.

final version (36 pages, 6 figures, 1 table)

References in corpus (32)

• 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
• Large deviations at level 2.5 for Markovian open quantum systems: quantum jumps and quantum state diffusion
• Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems
• Dynamical large deviations of reflected diffusions
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics
• Large deviations for Markov processes with stochastic resetting : analysis via the empirical density and flows or via excursions between resets
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Large deviations of currents in diffusions with reflective boundaries
• 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]
• Non-intersecting Brownian bridges in the flat-to-flat geometry
• Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models
• Inference of Markov models from trajectories via Large Deviations at Level 2.5 with applications to random walks in disordered media
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• 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
• Conditioning diffusion processes with killing rates
• Inhomogeneous asymmetric exclusion processes between two reservoirs : large deviations for the local empirical observables in the Mean-Field approximation

Cited by in corpus (7)

• Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
• Generalization of Fluctuation-Dissipation Theorem to Systems with Absorbing States
• First-passage time of a Brownian motion: two unexpected journeys
• Killing versus branching: Unexplored facets of diffusive relaxation
• Markov spin models for image generation : explicit large deviations with respect to the number of pixels
• Tamed Feynman-Kac diffusion processes: Killing-branching intertwine
• Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences