A Formal View on Level 2.5 Large Deviations and Fluctuation Relations

Abstract

We obtain the rate function for the level 2.5 of large deviations for pure jump and diffusion processes. This result is proved by two methods: tilting, for which a tilted process with an appropriate typical behavior is considered, and a spectral method, for which the scaled cumulant generating function is used. We also briefly discuss fluctuation relations, pointing out their connection with large deviations at the level 2.5.

Introduction to dynamical large deviations of Markov processes

“…24. [30] Derive the expression of the tilted generator L k for SDEs in which the noise is multiplicative, that is, in which the noise matrix σ depends on x. In this case, you must specify the stochastic convention used for interpreting the product σ(x)dW t .…”

Affinity- and topology-dependent bound on current fluctuations

“…Eq. (19) follows from an exact expression for the "level 2.5" large deviation function I(J , p) for the joint distribution of the fluctuating current J and the fluctuating density p = (p i ) [16,17]. By using the contraction principle, the large deviation function for the currents can be expressed as I(J ) = min p I(J , p) = I(J , p * (J )).…”

Level 2 and level 2.5 large deviation functionals for systems with and without detailed balance

“…In addition, the complete function ( ) J a characterizes the statistics of exponentially rare realizations a T that deviate substantially from á ñ a st . Recent applications of large deviation functions to describe rare events in statistical mechanics can be found in [8][9][10][11][12][13][14].Different Brownian functionals deriving from the same stochastic process ( ) x t have different large deviation functions. On the other hand, we may rewrite (1.1) as New J. Phys.…”

“…, it was dubbed level 2.5 [14]. For the above contraction, however, no closed form for NESS is known.…”

“…For the above contraction, however, no closed form for NESS is known. For the level 2.5 large deviation functionals, in contrast, closed expressions that are valid for equilibrium states and NESS, can be found in the mathematical and the mathematical physics literature [14,[19][20][21][22]. Closely related are the results of Bertini et al on large deviation functionals in macroscopic fluctuation theory [23,24].…”