About the $L^2$ analyticity of Markov operators on graphs

The local properties of the Markov processes of Ornstein-Uhlenbeck type

We prove the existence of a local time, the continuity of the local time about $t$, and the regular property for $a.e.$ $x\in R$ of a Ornstein-Uhlenbeck type $\{X_t,\ t\in R^+\}$ driven by a general Lévy process, under mild regularity conditions. We discuss the asymptotic behaviour of the local time when $X$ is ergodic. We also investigate the first passage problem. These results give precise information about the local properties of the sample functions.

Markov chains with heavy-tailed increments and asymptotically zero drift

We study the recurrence/transience phase transition for Markov chains on $\mathbb{R}_+$, $\mathbb{R}$, and $\mathbb{R}^2$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On $\mathbb{R}_+$, for example, we show that if the tail of the positive increments is about $c y^{-α}$ for an exponent $α\in (1,2)$ and if the drift at $x$ is about $b x^{-γ}$, then the critical regime has $γ= α-1$ and recurrence/transience is determined by the sign of $b + cπ\textrm{cosec} (πα)$. On $\mathbb{R}$…

About Gaussian filtering problems with general exponential quadratic criteria

Filtering problems with general exponential quadratic criteria are investigated for Gauss-Markov processes. In this setting, the Linear Exponential Gaussian and Risk-Sensitive filtering problems are solved and it is shown that they may have different solutions.

Some things we've learned (about Markov chain Monte Carlo)

This paper offers a personal review of some things we've learned about rates of convergence of Markov chains to their stationary distributions. The main topic is ways of speeding up diffusive behavior. It also points to open problems and how much more there is to do.

Weak invariance principle for the local times of partial sums of Markov Chains

Let X_{n} be an integer valued Markov Chain with finite state space. Let S_{n}=\sum_{k=0}^{n}X_{k} and let L_{n}(x) be the number of times S_{k} hits x up to step n. Define the normalized local time process t_{n}(x) by t_{n}(x)=\frac{L_{n}(\sqrt{n}(x)}{\sqrt{n}}. The subject of this paper is to prove a functional, weak invariance principle for the normalized sequence t_{n}, i.e. we prove that under some assumptions about the Markov Chain, the normalized local times converge in distribution to the local time of the Brownian Motion.

Álbum de figurinhas da Copa do Mundo: uma abordagem via cadeias de Markov

Using Markov chains, we present some probabilistic comments about the sticker album of 2014 Fifa World Cup.

A Markov chain approach to renormalization group transformations

We aim at an explicit characterization of the renormalized Hamiltonian after decimation transformation of a one-dimensional Ising-type Hamiltonian with a nearest-neighbor interaction and a magnetic field term. To facilitate a deeper understanding of the decimation effect, we translate the renormalization flow on the Ising Hamiltonian into a flow on the associated Markov chains through the Markov-Gibbs equivalence. Two different methods are used to verify the well-known conjecture that the eigenvalues of the linearization of this renormalization transformation about the fixed point bear importa…

Properties of centered random walks on locally compact groups and Lie groups

The basic aim of this paper is to study asymptotic properties of the convolution powers K^(n) = K * K * ... * K of a possibly non-symmetric probability density K on a locally compact, compactly generated group G. If K is centered, we show that the Markov operator T associated with K is analytic in L^p(G) for 1<p<\infty, and establish Davies-Gaffney estimates in L^2 for the iterated operators T^n. These results enable us to obtain various Gaussian bounds on K^(n). In particular, when G is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of…

广义主特征值与分支马尔可夫过程的全局生存性

Generalised principal eigenvalues and global survival of branching Markov processes

本研究将分支马尔可夫过程的全局生存性与椭圆算子广义主特征值关联，将相关概念推广至一般线性正算子半群，证明了广义主特征值及广义FKPP方程平稳解的新结论，并通过概率方法简化证明、构建反例。

What does a generic Markov operator look like

We consider generic i.e., forming an everywhere dense massive subset classes of Markov operators in the space $L^2(X,μ)$ with a finite continuous measure. Since there is a canonical correspondence that associates with each Markov operator a multivalued measure-preserving transformation (i.e., a polymorphism), as well as a stationary Markov chain, we can also speak about generic polymorphisms and generic Markov chains. The most important and inexpected generic properties of Markov operators (or Markov chains or polymorphisms) is nonmixing and totally nondeterministicity. It was not known even e…

马尔可夫群体模型平衡分布的界限

Bounding the Equilibrium Distribution of Markov Population Models

针对状态空间无限的连续时间马尔可夫链，提出一种通过几何边界和逐状态边界分析来获取平衡分布界限的新方法。

具有边界的分段确定性马尔可夫过程的数值方法

Numerical methods for piecewise deterministic Markov processes with boundary

本文研究了具有边界的分段确定性马尔可夫过程的数值方法，证明了广义柯尔莫戈洛夫方程在测试函数空间中的唯一解的存在性，分析了有限体积方案的正解，并通过不同传输控制协议窗口大小过程的模拟展示了该数值方法在描述具有边界的分段确定性马尔可夫过程密度演化中的有效性。

量子与经典马尔可夫链；精确可解的例子

Quantum vs classical Markov chains; Exactly solvable examples

本文提出图上可逆马尔可夫链的无硬币量子化方法，通过相似变换构造量子哈密顿量，并基于Askey方案中的正交多项式给出二十多个精确可解例子，以展示量子与经典马尔可夫链的对比。

Weak Markov Blankets in High-Dimensional, Sparsely-Coupled Random Dynamical Systems

This paper formulates a notion of high-dimensional random dynamical systems that couple to another system, like an embedding environment, in such a way that each system engages in controlled exchange with the other system. Using the adiabatic theorem and asymptotic arguments about the behaviour of systems with many degrees of freedom, we show that this sort of controlled, but not strictly sparse, coupling structure is ubiquitous in high-dimensional systems. In doing so we prove a conjecture of K Friston.