Stratonovich型可加泛函的集中性
Concentration of additive functionals of Stratonovich-type
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中文总结 AI 辅助
该研究针对Stratonovich型可加泛函的集中性问题，通过对称化Dirichlet型方法证明了其集中性不等式及方差上界，发现细致平衡下该泛函的集中性具有与Lebesgue型泛函不同的次高斯特性。
中文摘要 AI 辅助
Stratonovich型可加泛函$\u0305J_t=\frac{1}{t}\int_0^tU(X_s)\circ dX_s$近期在从个体涨落路径$(X_s)_{0\le s\le t}$的观测值$U$推断复杂系统热力学性质的研究中受到大量关注，其中$X_0$从某一一般测度初始化。尽管$\u0305J_t$的集中性结果是人们所期望的，但实际上几乎不存在相关结论。这些结果的证明比经典Lebesgue型泛函$\u03C1_t=\frac{1}{t}\int_0^t V(X_s)ds$的集中性结果的证明要困难得多，因为倾斜会改变Feynman-Kac生成元的整个二阶结构，而非仅作为加性势贡献，这使得即使在细致平衡条件下，生成元通常也非自伴。我们通过结合新有效势的对称化Dirichlet型来解决这一问题，该有效势将观测值与动力学的非平衡特性耦合起来。我们针对一般几何遍历扩散过程$X_s$的任意有界、足够光滑的向量值函数$U$，证明了$\u0305J_t$的集中性不等式，包括显式的次伽马型和Bernstein型不等式，并得到了${\rm Var}(\u0305J_t)$的显式上界。值得注意的是，在细致平衡条件下，$\u0305J_t$的集中性在所有时刻和所有偏差下均呈现明显的次高斯特性，其方差代理仅由噪声决定，且与谱间隙无关，这是$\u03C1_t$所没有的类似特性。
英文摘要
Additive functionals $\overline{J}_t=\frac{1}{t}\int_0^tU(X_s)\circ dX_s$ of Stratonovich-type recently attracted much attention in the context of inference of thermodynamic properties of complex systems from observations $U$ of individual fluctuating paths $(X_s)_{0\le s\le t}$, whereby $X_0$ is initiated from some general measure. Concentration results on $\overline{J}_t$, albeit desirable, are virtually nonexistent. They turn out to be significantly more challenging to prove than for classical Lebesgue-type functionals $\overlineρ_t=\frac{1}{t}\int_0^t V(X_s)ds$ because the tilt deforms the full second-order structure of the Feynman-Kac generator instead of contributing an additive potential. This renders the generator generally non-self-adjoint even under detailed balance. We overcome this by working with a symmetrized Dirichlet form with a new effective potential that now couples the observable to the non-equilibrium character of the dynamics. We prove concentration inequalities for $\overline{J}_t$ for any bounded, sufficiently smooth vector-valued function $U$ of a general geometrically ergodic diffusion process $X_s$, including explicit sub-gamma and Bernstein-type inequalities, and we obtain explicit upper bounds on ${\rm Var}(\overline{J}_t)$. Strikingly, under detailed balance the concentration of $\overline{J}_t$ is distinctively sub-Gaussian at all times and all deviations, with a variance proxy fixed by the noise alone and independent of the spectral gap, which has no analog for $\overlineρ_t$.
