Fluctuations of dynamical observables in linear diffusions with time delay: a Riccati-based approach

TL;DR

This paper develops a Riccati-based mathematical framework to analyze fluctuations of time-integrated observables in linear diffusions with delay, addressing a key challenge in non-Markovian process analysis.

Contribution

It introduces a Markovian embedding and Riccati differential equations approach to compute large deviation functions in delayed linear diffusions, a novel method in this context.

Findings

Explicit formulas for scaled cumulant generating functions derived

Identification of fixed points for long-time behavior of fluctuations

Analysis of fluctuation relations at domain boundaries

Abstract

Our current understanding of fluctuations of dynamical (time-integrated) observables in non- Markovian processes is still very limited. A major obstacle is the lack of an appropriate theoretical framework to evaluate the associated large deviation functions. In this paper we bypass this difficulty in the case of linear diffusions with time delay by using a Markovian embedding procedure that introduces an infinite set of coupled differential equations. We then show that the generating functions of current-type observables can be computed at arbitrary finite time by solving matrix Riccati differential equations (RDEs) somewhat similar to those encountered in optimal control and filtering problems. By exploring in detail the properties of these RDEs and of the corresponding continuous-time algebraic Riccati equations (CAREs), we identify the generic fixed point towards which the solutions converge in the long-time limit. This allows us to derive the explicit expressions of the scaled cumulant generating function (SCGF), of the pre-exponential factors, and of the effective (or driven) process that describes how fluctuations are created dynamically. Finally, we describe the special behavior occurring at the limits of the domain of existence of the SCGF, in connection with fluctuation relations for the heat and the entropy production.