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When analysing statistical systems or stochastic processes, it is ofteninteresting to ask how they behave given that some observable takes someprescribed value. This conditioning problem is well understood within thelinear operator formalism based on rate matrices or Fokker-Planck operators,which describes the dynamics of many independent random walkers. Relying oncertain spectral properties of the biased linear operators, guaranteed by thePerron-Frobenius theorem, an effective process can be found such that its pathprobability is equivalent to the conditional path probability. In this paper,we extend those results for nonlinear Markov processes that appear when themany random walkers are no longer independent, and which can be describednaturally through a Lagrangian-Hamiltonian formalism within the theory of largedeviations at large volume. We identify the appropriate spectral problem asbeing a Hamilton-Jacobi equation for a biased Hamiltonian, for which weconjecture that two special global solutions exist, replacing thePerron-Frobenius theorem concerning the positivity of the dominant eigenvector.We then devise a rectification procedure based on a canonical gaugetransformation of the biased Hamiltonian, yielding an effective dynamics inagreement with the original conditioning. Along the way, we present simpleexamples in support of our conjecture, we examine its consequences on importantphysical objects such as the fluctuation symmetries of the biased and rectifiedprocesses as well as the dual dynamics obtained through time-reversal. We applyall those results to simple independent and interacting models, including astochastic chemical reaction network and a population process called theBrownian Donkey.

Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume