论文标题
部分可观测时空混沌系统的无模型预测
Second Order Bismut formulae and applications to Neumann semigroups on manifolds
论文作者
论文摘要
让$ m $成为带有边界$ \ partial m $的完整连接的riemannian歧管，让$ p_t $为$ \ frac {1} {1} {2} l $ $ c^1 $ c^1 $ -vector field $ z $ z $ on $ m $。我们为$ lp_t f $和$ {\ rm hess} _ {p_tf} $建立了bismut类型公式，并在适当的曲率条件下对这些数量的估计值进行了当前的估计。如果$ p_t $在$ l^2（μ）中为某些概率度量$μ$对称时，建立了一种新型的log-sobolev不平等现象，将相对熵$ h $，stein差异$ s $和相对渔民信息链接起来，以及相对的渔民信息$ i $ $ $ $ $ $ $在没有边界的情况下，将其概括为作者的最新工作。
Let $M$ be a complete connected Riemannian manifold with boundary $\partial M$, and let $P_t$ be the Neumann semigroup generated by $\frac{ 1}{ 2} L$ where $L=Δ+Z$ for a $C^1$-vector field $Z$ on $M$. We establish Bismut type formulae for $LP_t f$ and ${\rm Hess}_{P_tf}$ and present estimates of these quantities under suitable curvature conditions. In case when $P_t$ is symmetric in $L^2(μ)$ for some probability measure $μ$, a new type of log-Sobolev inequality is established which links the relative entropy $H$, the Stein discrepancy $S$, and relative Fisher information $I$, generalizing the authors' recent work in the case without boundary.
