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Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts

Cecile Monthus

Abstract

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension $d$ involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing $b$ ; (ii) discrete-time random walks in continuous space with a small time-step $τ$. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits $b \to 0$ and $τ\to 0$ respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits $b \to 0$ and $τ\to 0$ for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.

Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts

Abstract

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing ; (ii) discrete-time random walks in continuous space with a small time-step . One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits and respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits and for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.
Paper Structure (63 sections, 230 equations)

This paper contains 63 sections, 230 equations.

Table of Contents

  1. Introduction
  2. Large deviations at various levels for Markov processes
  3. Goals and organization of the present paper
  4. Large deviations at level 2.5 for Markov chains and for diffusions
  5. Reminder on large deviations for discrete-time Markov Chain in continuous-space in dimension $d$
  6. Probability of a trajectory $x(0 \leq t \leq T)$ in terms of its time-empirical 2-point density $\rho^{(2)}(\vec{x}, \vec{y})$ with its constraints
  7. Number ${\cal N}^{[2.5]}_T\left[ \rho^{(2)}(.,.) \right]$ of trajectories $\vec{x}(0 \leq t \leq T)$ of duration $T$ with a given empirical 2-point density $\rho^{(2)}(.,.)$
  8. Large deviations at level 2.5 for the probability distribution $P_T^{[2.5]} [ \rho^{(2)}(.,.) ]$ of the empirical 2-point density $\rho^{(2)}(.,.)$
  9. Trajectory observables that can be rewritten in terms of the empirical 2-point density $\rho^{(2)}(.,.)$
  10. Reminder on large deviations for continuous-time Markov jump processes in discrete space
  11. Probability of a trajectory $x(0 \leq t \leq T)$ in terms of its empirical probability $P(.)$ and its empirical flows $Q(.,.)$
  12. Replacing the empirical flows $Q(.,.)$ by the empirical currents $J(.,.)$ and the empirical activities $A(.,.)$
  13. Number ${\cal N}^{[2.5]}_T\left[ P(.) ; Q(.,.) \right]$ of trajectories of duration $T$ with given empirical observables $\left[ P(.) ; Q(.,.) \right]$
  14. Joint probability distribution $P^{[2.5]}_T \left[ P(.) ; Q(.,.) \right]$ of the empirical observables $\left[ P(.) ; Q(.,.) \right]$
  15. Explicit level 2.25 for the joint distribution of the empirical probability $P(.)$ and the empirical currents $J(.,.)$
  16. ...and 48 more sections