Large Deviations of the Entropy Production Rate for a Class of Gaussian Processes

We prove a large deviation principle (LDP) and a fluctuation theorem (FT) for the entropy production rate (EPR) of the following $d$ dimensional stochastic differential equation \begin{equation*} d X_{t}=AX_{t} d t+\sqrt{Q} d B_{t} \end{equation*} where $A$ is a real normal stable matrix, $Q$ is positive definite, and the matrices $A$ and $Q$ commute. The rate function for the EPR takes the following explicit form: \begin{equation*} I(x)=\left\{ \begin{array}{ll} x\frac{\sqrt{1+\ell_0(x)}-1}{2}+\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{\alpha_k^2- \beta_k^2\ell_0(x)}+\alpha_k\right) , & x\ge 0 , \\ -x\frac{\sqrt{1+\ell_0(x)}+1}{2} +\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{\alpha_k^2- \beta_k^2\ell_0(x)}+\alpha_k\right) , &x<0, \end{array} \right. \end{equation*} where $\alpha_{k}\pm {\rm i} \beta_{k}$ are the eigenvalues of $A$, and $\ell_0(x)$ is the unique solution of the equation: \begin{align*} |x|={\sqrt{1+\ell}} \times \sum_{k=1}^{d} \frac{\beta_k^2}{\sqrt{\alpha_k^2 -\ell\beta_k^2} },\qquad -1 \le \ell< \min_{k=1,...,d}\{\frac{\alpha_k^2}{\beta_k^2}\}. \end{align*} Simple closed form formulas for rate functions are rare and our work identifies an important class of large deviation problems where such formulas are available. The logarithmic moment generating function (the fluctuation function) $\Lambda$ associated with the LDP has a closed form (see it in the paper). The functions $\Lambda(\lambda)$ and $I(x)$ satisfy the Cohen-Gallavotti symmetry properties. In particular, the functions $I$ and $\Lambda$ do not depend on the diffusion matrix $Q$, and are determined completely by the real and imaginary parts of the eigenvalues of $A$. Formally, the deterministic system with $Q=0$ has zero EPR and thus the model exhibits a phase transition in that the EPR changes discontinuously at $Q = 0$.