We prove a large deviation principle (LDP) and a fluctuation theorem for the entropy production rate (EPR) of the following d dimensional stochastic differential equation $\mathrm{d}{X}_t=A{X}_t\mathrm{d}t+\sqrt{Q}\mathrm{d}{B}_t$, 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: $I(x)=x\frac{\sqrt{1+{\ell }_0(x)}-1}{2}+\frac{1}{2}\sum _{k=1}^d\left(\sqrt{{\alpha }_k^2-{\beta }_k^2{\ell }_0(x)}+{\alpha }_k\right)$ for x ≥ 0 and $I(x)=-x\frac{\sqrt{1+{\ell }_0(x)}+1}{2}+\frac{1}{2}\sum _{k=1}^d\left(\sqrt{{\alpha }_k^2-{\beta }_k^2{\ell }_0(x)}+{\alpha }_k\right)$ for x < 0, where α_k ±iβ_k are the eigenvalues of A and ℓ₀(x) is the unique solution of the equation $\left|x\right|=\sqrt{1+\ell }\times {\sum }_{k=1}^d\frac{{\beta }_k^2}{\sqrt{{\alpha }_k^2-\ell {\beta }_k^2}},-1\le \ell <{\min }_{k=1,\dots ,d}\left\{\frac{{\alpha }_k^2}{{\beta }_k^2}\right\}$. 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) Λ associated with the LDP is given as $\mathrm{\Lambda }(\lambda )=-\frac{1}{2}\sum _{k=1}^d\left(\sqrt{{\alpha }_k^2-4\lambda (1+\lambda ){\beta }_k^2}+{\alpha }_k\right)$ for $\lambda \in \mathcal{D}$ and Λ(λ) = ∞ for $\lambda \notin \mathcal{D}$, where $\mathcal{D}$ is the domain of Λ. The functions Λ(λ) and I(x) satisfy the Cohen–Gallavotti symmetry properties: $\mathrm{\Lambda }(x)=\mathrm{\Lambda }(-(1+x)),I(x)=I(-x)-x,\text{\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}x\in \mathbb{R}$. In particular, the functions I and Λ 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.

中文翻译：

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一类高斯过程的熵产生率的大偏差

我们证明了以下d维随机微分方程的大偏差原理（LDP）和波动定理为熵产生率（EPR）$\mathrm{d}{X}_{吨}=\mathrm{一种}{X}_{吨}\mathrm{d}吨+\sqrt{问}\mathrm{d}{乙}_{吨}$，其中A是实正规稳定矩阵，Q是正定矩阵，矩阵A和Q交换。EPR 的速率函数采用以下显式形式：$\mathrm{一世}(X)=X\frac{\sqrt{1+{\ell }_0(X)}-1}{2}+\frac{1}{2}\sum _{克=1}^d\left(\sqrt{{\alpha }_{克}^2-{\beta }_{克}^2{\ell }_0(X)}+{\alpha }_{克}\right)$对于x ≥ 0 和$\mathrm{一世}(X)=-X\frac{\sqrt{1+{\ell }_0(X)}+1}{2}+\frac{1}{2}\sum _{克=1}^d\left(\sqrt{{\alpha }_{克}^2-{\beta }_{克}^2{\ell }_0(X)}+{\alpha }_{克}\right)$为X <0，其中α _ķ ±I β _ķ是的本征值阿和ℓ ₀（X）是方程的独特的解决方案$\left|X\right|=\sqrt{1+\ell }\times {\sum }_{克=1}^d\frac{{\beta }_{克}^2}{\sqrt{{\alpha }_{克}^2-\ell {\beta }_{克}^2}},-1\le \ell <{\mathrm{分钟}}_{克=1,\dots ,d}\left\{\frac{{\alpha }_{克}^2}{{\beta }_{克}^2}\right\}$. 速率函数的简单封闭式公式很少见，我们的工作确定了一类重要的大偏差问题，其中此类公式可用。与 LDP 相关的对数矩生成函数（波动函数）Λ 给出为$\mathrm{\Lambda }(\lambda )=-\frac{1}{2}\sum _{克=1}^d\left(\sqrt{{\alpha }_{克}^2-4\lambda (1+\lambda ){\beta }_{克}^2}+{\alpha }_{克}\right)$ 为了 $\lambda \in \mathcal{D}$并且 Λ( λ ) = ∞对于$\lambda \notin \mathcal{D}$， 在哪里 $\mathcal{D}$是Λ的域。函数 Λ( λ ) 和I ( x ) 满足 Cohen-Gallavotti 对称性质：$\mathrm{\Lambda }(X)=\mathrm{\Lambda }(-(1+X)),\mathrm{一世}(X)=\mathrm{一世}(-X)-X,\text{\hspace{0.17em}对所有人\hspace{0.17em}}X\in \mathbb{电阻}$. 特别是，函数I和 Λ 不依赖于扩散矩阵Q并且完全由A的特征值的实部和虚部确定。形式上，Q = 0的确定性系统具有零 EPR，因此，该模型表现出相变，即 EPR 在Q = 0处不连续变化。