Large deviations for quadratic functionals of stable Gauss–Markov chains and entropy production

Abstract

In this paper, we establish a large deviation principle for the entropy production rate of possible non-stationary, centered stable Gauss–Markov chains, verifying the Gallavotti–Cohen symmetry. We reach this goal by developing a large deviation theory for quasi-Toeplitz quadratic functionals of multivariate centered stable Gauss–Markov chains, which differ from a perfect Toeplitz form by the addition of quadratic boundary terms.

Work fluctuations for a confined Brownian particle: the role of initial conditions

“…which weighs each trajectory realisation combining the distribution of the initial values with the Onsager-Machlup weight [40]. To compute the rate function I(w), we follow the approach developed in [38], which is briefly summarised in the next subsection. Application of this approach to our problem is made in section 3.…”

“…We mention that an alternative approach to deal with the work injected by a Gaussian external force into an underdamped Brownian particle is suggested in [28,30,31]. Although both methods are ultimately traced back to Gaussian integrals for computational purposes, the theory developed in [38] provides, with mathematical rigour, a large deviation principle for a general class of quadratic functionals and a comprehensive recipe for evaluating their rate functions.…”

“…In order to evaluate the rate function I(w), we resort to the approach recently proposed in [38] for quadratic functionals of stable Gauss-Markov chains, whose applicability to our problem is feasible upon a convenient time discretization. Results for the original problem are later obtained by performing a continuum limit.…”

“…be a quadratic functional of the chain, where ⟨•, •⟩ is the Euclidean scalar product and L, U, R, and V are d × d real matrices with L, U, and R symmetric. With full probability, the typical value of W N is given by the following law of large numbers [38] lim…”

“…In section 2 we present the model and introduce the work injected by the random force. In this section, we also report a brief summary of the analytical approach developed in [38], which we use to compute the rate function. Section 3 is devoted to the computation of the scaled cumulant generating function (SCGF) and the rate function, the latter being the Legendre transform of the former.…”