Invariant Measure for Stochastic Functional Differential Equations in Hilbert Spaces

Classical Solutions of Path-dependent PDEs and Functional Forward-Backward Stochastic Systems

In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Itô calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.

希尔伯特空间中的随机最优控制：值函数的$C^{1,1}$正则性与基于粘性解的最优综合

Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions

本文用动态规划方法研究无穷维随机最优控制，证明值函数的$C^{1,1}$正则性，并基于$B$-连续粘性解构造最优反馈控制，应用于随机反应-扩散和延迟方程。

The splitting lemmas for nonsmooth functionals on Hilbert spaces II. The case at infinity

We generalize the Bartsch-Li's splitting lemma at infinity for $C^2$-functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow methods our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [9] with some techniques from [11], [17], [18]. A simple application is also presented.

具有完全单调核的随机Volterra积分方程与在Hilbert空间上的随机微分方程的最优控制

Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators

本文研究了具有完全单调核的随机Volterra积分方程的最优控制问题，通过马尔可夫提升方法将其转化为Hilbert空间上的随机微分方程问题，并证明了新的平滑性质，从而建立了Hamilton-Jacobi-Bellman方程的弱解存在性和唯一性，以及构造了最优反馈控制。

在希尔伯特空间中相互作用粒子系统的随机最优控制及其应用

Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications

本文研究了在希尔伯特空间中相互作用粒子系统的随机最优控制问题，建立了均场哈密尔顿-雅可比-贝尔曼方程的粘性解理论，并将其应用于经济学中的随机延迟微分方程和随机偏微分方程问题。

可分希尔伯特空间中的Schauder正则性结果

Schauder regularity results in separable Hilbert spaces

本文在可分希尔伯特空间中，针对与半线性随机偏微分方程相关的转移半群弱生成元驱动的平稳与演化方程，证明其解的Schauder型正则性估计。

On critical normal sections for two-dimensional immersions in R^n and a Riemann-Hilbert problem

For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvature of the normal bundle is provided.

论一般希尔伯特空间中的波类微分方程：作为工程科学应用的梁的欧拉-伯努利弯曲振动的泛函分析研究

On Wave-Like Differential Equations in General Hilbert Space. The Functional Analytic Investigation of Euler-Bernoulli Bending Vibrations of a Beam as an Application in Engineering Science

本文在一般希尔伯特空间泛函分析框架下研究波类微分方程，重点严格构造并分析了一维弹性梁欧拉-伯努利弯曲振动在不同边界条件下的四阶正自伴微分算子，并与弦的自由波动进行了比较。

A Hilbert space approach to fractional difference equations

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator $1 - τ^{-1}$ with the right shift $τ^{-1}$ on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.

The splitting lemmas for nonsmooth functionals on Hilbert spaces I

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_Ωf(x, u,..., D^mu)dx$ as in (\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower…

Yamabe flow and ADM Mass on asymptotically flat manifolds

In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow is the gradient flow of Einstein-Hilbert functional on asymptotically flat manifolds