Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

A detailed proof of the von Neumann's Quantum Ergodic Theorem

We present a simplified proof of the von Neumann's Quantum Ergodic Theorem. This important result was initially published in german by J. von Neumann in 1929. We are interested here in the time evolution $ψ_t$, $t\geq 0$, (for large times) under the Schrodinger equation associated to a given fixed Hamiltonian $H : \mathcal{H} \to \mathcal{H}$ and a general initial condition $ψ_0$. The dimension of the Hilbert space $\mathcal{H}$ is finite.

von Neumann代数上的KMS Dirichlet形式、强制性与超有界Markov半群

KMS Dirichlet forms, coercivity and super-bounded Markovian semigroups on von Neumann algebras

本文在von Neumann代数上构造了与Araki模哈密顿量特征值相关的KMS Dirichlet形式，证明了其强制性与谱增长，并引入超有界性概念，最终应用于量子Ornstein-Uhlenbeck半群的构造。

马尔可夫开放量子系统中的平衡态趋近

Approach to equilibrium in Markovian open quantum systems

该研究针对由 von Neumann-Lindblad 方程描述的马尔可夫开放量子系统，在量子细致平衡条件下证明无限自由度情形的回归平衡性质。

Markov semigroups with hypocoercive-type generator in Infinite Dimensions: Ergodicity and Smoothing

We start by considering infinite dimensional Markovian dynamics in R^m generated by operators of hypocoercive type and for such models we obtain short and long time pointwise estimates for all the derivatives, of any order and in any direction, along the semigroup. We then look at infinite dimensional models (in (Rm)^{Z ^d}) produced by the interaction of infinitely many finite dimensional dissipative dynamics of the type indicated above. For these infinite dimensional models we study finite speed of propagation of information, well-posedness of the semigroup, time behaviour of the derivatives…

有限 von Neumann 代数上的遍历量子过程

Ergodic Quantum Processes on Finite von Neumann Algebras

本文研究有限 von Neumann 代数上由遍历变换驱动的随机正线性算子复合过程，证明其在合理假设下几乎必然指数坍缩为替换信道，并推广了 [MS22] 的定理至无限维情形。

Ergodic theory for SDEs with extrinsic memory

We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob--Khas'minskii theorem. The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition o…