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Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?

Cecile Monthus

Abstract

In the field of large deviations for stochastic dynamics, the canonical conditioning of a given Markov process with respect to a given time-local trajectory observable over a large time-window has attracted a lot of interest recently. In the present paper, we analyze the following inverse problem: when two Markov generators are given, is it possible to connect them via some canonical conditioning and to construct the corresponding time-local trajectory observable? We focus on continuous-time Markov processes and obtain the following necessary and sufficient conditions: (i) for continuous-time Markov jump processes, the two generators should involve the same possible elementary jumps in configuration space, i.e. only the values of the corresponding rates can differ; (ii) for diffusion processes, the two Fokker-Planck generators should involve the same diffusion coefficients, i.e. only the two forces can differ. In both settings, we then construct explicitly the various time-local trajectory observables that can be used to connect the two given generators via canonical conditioning. This general framework is illustrated with various applications involving a single particle or many-body spin models. In particular, we describe several examples to show how non-equilibrium Markov processes with non-vanishing steady currents can be interpreted as the canonical conditionings of detailed-balance processes with respect to explicit time-local trajectory observables.

Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?

Abstract

In the field of large deviations for stochastic dynamics, the canonical conditioning of a given Markov process with respect to a given time-local trajectory observable over a large time-window has attracted a lot of interest recently. In the present paper, we analyze the following inverse problem: when two Markov generators are given, is it possible to connect them via some canonical conditioning and to construct the corresponding time-local trajectory observable? We focus on continuous-time Markov processes and obtain the following necessary and sufficient conditions: (i) for continuous-time Markov jump processes, the two generators should involve the same possible elementary jumps in configuration space, i.e. only the values of the corresponding rates can differ; (ii) for diffusion processes, the two Fokker-Planck generators should involve the same diffusion coefficients, i.e. only the two forces can differ. In both settings, we then construct explicitly the various time-local trajectory observables that can be used to connect the two given generators via canonical conditioning. This general framework is illustrated with various applications involving a single particle or many-body spin models. In particular, we describe several examples to show how non-equilibrium Markov processes with non-vanishing steady currents can be interpreted as the canonical conditionings of detailed-balance processes with respect to explicit time-local trajectory observables.
Paper Structure (58 sections, 211 equations)

This paper contains 58 sections, 211 equations.

Table of Contents

  1. Introduction
  2. Reminder on the canonical conditioning for Markov jump processes
  3. Properties of the Markov generator $w(.,.)$ with its steady state $P_*(.)$
  4. Spontaneous fluctuations of time-averaged observables over a large-time window $[0,T]$
  5. Empirical probability $P(.)$ and empirical flows $Q(.,.)$ with their constitutive constraints
  6. Explicit joint distribution of the empirical probability $P(.)$ and the empirical flows $Q(.,.)$ for large $T$
  7. Different perspective when the empirical flows $Q(.,.)$ are replaced by the empirical Markov generator $w^E(.,.)$
  8. Generating function of the empirical probability $P(.)$ and the empirical flows $Q(.,.)$
  9. Consequences of the constraints on the empirical observables $[P(.);Q(.,.)]$ for their conjugated variables $[\omega(.); \lambda(.,.)]$
  10. Eigenvalue problem governing the generating function $Z_T^{[\omega(.); \lambda(.,.)]}(x \vert x_0)$ for large $T$
  11. Functional Legendre transform between the explicit rate function $I_{2.5} [ P(.), Q(.,.)]$ and the eigenvalues $G[\omega(.);\lambda(.,.)]$
  12. Canonical conditionings with respect to the empirical probability $P(.)$ and to the empirical flows $Q(.,.)$
  13. Conditioned Markov generator $w^{Cond[\omega(.); \lambda(.,.)]}(.,.)$ associated to the deformed-generator $w^{[\omega(.); \lambda(.,.)]}(.,.)$
  14. Special cases $[\omega(.); \lambda(.,.)]$ with vanishing eigenvalue $G[\omega(.); \lambda(.,.)] =0$ and trivial left eigenvector $l^{[\omega(.); \lambda(.,.)]}(x)=1$
  15. Inverse problem for the conditioning of Markov jump processes
  16. ...and 43 more sections