Yves Le Jan
Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central series of the fundamental group.
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In the investigation of limits of Markov chains, the presence of states which become instantaneous states in the limit may prevent the convergence of the chain in the Skorohod topology. We present in this article a weaker topology adapted to handle this situation. We use this topology to derive the limit of random walks among random traps and sticky zero-range processes.
Since the work of Lawler and Werner on "loop soups", these ensembles have also been the object of many investigations. Their properties can be studied in the context of rather general Markov processes, in particular Markov chains on graphs. The purpose of the present work is to explore their topological properties.
The Markov Accessibility Topology and its Probabilistic Structure
本文将典范拓扑与离散时间马尔可夫链的转移算子关联，定义了马尔可夫可达拓扑，建立了概率可达性与序理论拓扑的联系，还刻画了吸收子集等概念并研究了通信商与可 lump 性的关系。
The goal of this paper is to simplify and strengthen the Le Jan-Qian approximation scheme of studying the uniqueness of signature problem to the non-Markov setting. We establish a general framework for a class of multidimensional stochastic processes over [0,1] under which with probability one, the signature (the collection of iterated path integrals in the sense of rough paths) is well-defined and determines the sample paths of the process up to reparametrization. In particular, by using the Malliavin calculus we show that our method applies to a class of Gaussian processes including fraction…
The topology of weak convergence does not account for the growth of information over time that is captured in the filtration of an adapted stochastic process. For example, two adapted stochastic processes can have very similar laws but give completely different results in applications such as optimal stopping, queuing theory, or stochastic programming. To address such discontinuities, Aldous introduced the extended weak topology, and subsequently, Hoover and Keisler showed that both, weak topology and extended weak topology, are just the first two topologies in a sequence of topologies that ge…
Wireless Network Topology Inference: A Markov Chains Approach
本文提出一种基于马尔可夫链匿名状态观测的无线网络拓扑推断方法，构造一致估计量逼近转移矩阵，实验表明其在高拥塞条件下优于传递熵。
Universal approximation by signatures for infinite-dimensional rough paths
本文提出基于无限维粗糙路径的通用逼近定理，用于随机偏微分方程解的量的逼近，并为远期利率定价提供新方法。
We study a family of measures originating from the signatures of the irreducible components of representations of the unitary group, as the size of the group goes to infinity. Given a random signature $λ$ of length $N$ with counting measure $\mathbf{m}$, we obtain a random signature $μ$ of length $N-1$ through projection onto a unitary group of lower dimension. The signature $μ$ interlaces with the signature $λ$, and we record the data of $μ,λ$ in a random rectangular Young diagram $w$. We show that under a certain set of conditions on $λ$, both $\mathbf{m}$ and $w$ converge as $N\to\infty$. W…
The main purpose of this paper is to show that Markov solutions to the 3D Navier--Stokes equations driven by Gaussian noise have the strong Feller property up to the critical topology given by the domain of the Stokes operator to the power one-fourth.
Ising Disks: Topology Preserving Glauber Dynamics
本文提出一种动态模型，其状态空间为可缩立方体集，通过局部可决定的转移规则模拟拓扑约束下的Eden模型，并证明在平面情况下状态空间连通，且在 fugacity 小于阈值时马尔可夫链具有遍历性。
Further remarks on absorbing Markov decision processes
该研究基于已有成果推导了吸收马尔可夫决策过程的占用测度空间在弱拓扑与弱-强拓扑下收敛序列一致的结论，通过反例说明条件不可替代且结论不适用于边缘分布空间，同时澄清了过往文献的相关结果。
A careful look at rough path topology applied to Brownian motion reveals new possible properties of the well-known Lévy area, in particular the presence of an intrinsic drift of this area. Using renormalization limit of Markov chains on periodic graphs, we present a construction of such a non-trivial drift and give an explicit formula for it. Several examples for which explicit computations are made are included.
We introduce two models for multi-type random trees motivated by studies of trait dependence in the evolution of species. Our discrete time model, the multi-type ERM tree, is a generalization of Markov propagation models on a random tree generated by a binary search or `equal rates Markov' mechanism. Our continuous time model, the multi-type Yule tree with mutations, is a multi-type generalization of the tree generated by a pure birth or Yule process. We study type dependent topological properties of these two random tree models. We derive asymptotic results that allow one to infer model param…
Convergence to the Brownian CRT for critical branching Markov processe
证明有限方差临界分支过程在重标度下族谱树收敛到布朗连续随机树，建立普适标度极限。
The adapted weak topology is an extension of the weak topology for stochastic processes designed to adequately capture properties of underlying filtrations. With the recent work of Bart--Beiglböck-P. as starting point, the purpose of this note is to recover with topological arguments the intriguing result by Backhoff-Bartl-Beiglböck-Eder that all adapted topologies in discrete time coincide. We also derive new characterizations of this topology including descriptions of its trace on the sets of Markov processes and processes equipped with their natural filtration. To emphasize the generality o…
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
Operator semigroups in the mixed topology and the infinitesimal description of Markov processes
本研究在装备混合拓扑的连续函数空间上定义了非线性$C_0$-半群，刻画了其线性情形的生成元与积分表示，证明了马尔可夫转移半群的无穷小可重构性，引入马尔可夫核算子并给出唯一性判据，还将凸半群与最优控制值函数相关联。
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