Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume

Abstract

When analysing statistical systems or stochastic processes, it is often interesting to ask how they behave given that some observable takes some prescribed value. This conditioning problem is well understood within the linear operator formalism based on rate matrices or Fokker-Planck operators, which describes the dynamics of many independent random walkers. Relying on certain spectral properties of the biased linear operators, guaranteed by the Perron-Frobenius theorem, an effective process can be found such that its path probability is equivalent to the conditional path probability. In this paper, we extend those results for nonlinear Markov processes that appear when the many random walkers are no longer independent, and which can be described naturally through a Lagrangian-Hamiltonian formalism within the theory of large deviations at large volume. We identify the appropriate spectral problem as being a Hamilton-Jacobi equation for a biased Hamiltonian, for which we conjecture that two special global solutions exist, replacing the Perron-Frobenius theorem concerning the positivity of the dominant eigenvector. We then devise a rectification procedure based on a canonical gauge transformation of the biased Hamiltonian, yielding an effective dynamics in agreement with the original conditioning. Along the way, we present simple examples in support of our conjecture, we examine its consequences on important physical objects such as the fluctuation symmetries of the biased and rectified processes as well as the dual dynamics obtained through time-reversal. We apply all those results to simple independent and interacting models, including a stochastic chemical reaction network and a population process called the Brownian Donkey.

arXiv comment61 pages, 18 figures, authors are ordered in alphabetic order

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• Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator
• Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels
• Markov trajectories : Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• Conditioning diffusion processes with respect to the local time at the origin
• Dynamical equivalence classes for Markov jump processes
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
• Transport properties of diffusive particles conditioned to survive in trapping environments
• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?
• Markov spin models for image generation : explicit large deviations with respect to the number of pixels
• Conditioning two diffusion processes with respect to their first-encounter properties