Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume

TL;DR

This paper extends the theory of conditioned Markov processes to nonlinear cases using a Hamiltonian-Lagrangian formalism, proposing a spectral problem as a Hamilton-Jacobi equation and demonstrating its application to physical models.

Contribution

It introduces a novel approach to analyze nonlinear Markov processes conditioned on observables using a Hamilton-Jacobi spectral problem and a gauge transformation for effective dynamics.

Findings

Spectral problem identified as a Hamilton-Jacobi equation for biased Hamiltonian

Conjecture of two global solutions replacing Perron-Frobenius theorem

Effective dynamics consistent with original conditioning through gauge transformation

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.