非退化扩散中熵产生的大时间极限与消失噪声极限
The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions
AI总结：
本文研究非平衡非退化扩散过程中熵产生泛函在大时间和消失噪声极限下的渐近行为，证明其满足不依赖边界项的大数定律与具有Gallavotti-Cohen对称性的局部大偏差原理，并将问题归结为临界点附近生成元形变的二次近似首特征值分析。
摘要：
我们研究一族与形如 $\mathrm{d} X_s = -\nabla V(X_s) \\, \mathrm{d} s + b(X_s) \\, \mathrm{d} s + \sqrt{2ε} \\, \mathrm{d} W_s$ 的随机微分方程相关联的熵产生泛函的行为，其中 $b$ 是一个全局 Lipschitz 的非保守向量场，使系统处于非平衡状态，重点关注大时间极限以及随后的消失噪声极限。该族的不同成员对应于边界项的不同选择。我们的分析给出了一个大数定律和一个局部大偏差原理，该原理不依赖于边界项的选择，并展现出 Gallavotti--Cohen 对称性。我们使用来自半群理论和半经典分析的技术，将泛函渐近行为的描述归结为研究无穷小生成元在 $V$ 的临界点附近的形变的二次近似的首特征值。
英文摘要：
We investigate the behaviour of a family of entropy production functionals associated to stochastic differential equations of the form $\mathrm{d} X_s = -\nabla V(X_s) \, \mathrm{d} s + b(X_s) \, \mathrm{d} s + \sqrt{2ε} \, \mathrm{d} W_s $, where $b$ is a globally Lipschitz nonconservative vector field keeping the system out of equilibrium, with emphasis on the large-time limit and then the vanishing-noise limit. Different members of the family correspond to different choices of boundary terms. Our analysis yields a law of large numbers and a local large deviation principle which does not depend on the choice of boundary terms and which exhibits a Gallavotti--Cohen symmetry. We use techniques from the theory of semigroups and from semiclassical analysis to reduce the description of the asymptotic behaviour of the functional to the study of the leading eigenvalue of a quadratic approximation of a deformation of the infinitesimal generator near critical points of $V$.
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