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

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, we identify the generic fixed point toward 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 preexponential 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.