On the Dirichlet problem for asymmetric zero-range process on increasing domains

Real zeros of random Dirichlet series

Let $F(σ)$ be the random Dirichlet series $F(σ)=\sum_{p\in\mathcal{P}} \frac{X_p}{p^σ}$, where $\mathcal{P}$ is an increasing sequence of positive real numbers and $(X_p)_{p\in\mathcal{P}}$ is a sequence of i.i.d. random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=1/2$. We prove that, for certain conditions on $\mathcal{P}$, if $\sum_{p\in\mathcal{P}}\frac{1}{p}<\infty$ then with positive probability $F(σ)$ has no real zeros while if $\sum_{p\in\mathcal{P}}\frac{1}{p}=\infty$, almost surely $F(σ)$ has an infinite number of real zeros.

Malliavin and dirichlet structures for independent random variables

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron-Stein inequality can be interpreted as a Poincar{é} inequality or that Hoeffding decomposition of U-statistics can be interpreted as a chaos decomposition. We obtain a version of the Lyapounov central limit theorem for independent …

Degenerate Dirichlet Problems Related to the Ergodic Property of an Elasto-Plastic Oscillator Excited by a Filtered White Noise

A stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic properties of the process and characterize the corresponding invariant measure. This extends Bensoussan-Turi's method (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008) with a significant additional difficulty of increasing the dimension. Two points boundary value problem in dimension 1 is replaced by elliptic equations in dimension 2. In the present context, Khasminskii's method (Stochastic Stabilit…

带长跳跃和无穷延伸储库的WASEP的平稳涨落

Stationary fluctuations for the WASEP with long jumps and infinitely extended reservoirs

该研究针对带长跳跃与无穷延伸储库的弱非对称排斥过程，证明其平稳涨落依动力学非对称强度的不同，分别由广义Ornstein-Uhlenbeck过程或带Dirichlet边界条件的随机Burgers方程支配。

SINR in wireless networks and the two-parameter Poisson-Dirichlet process

Stochastic geometry models of wireless networks based on Poisson point processes are increasingly being developed with a focus on studying various signal-to-interference-plus-noise ratio (SINR) values. We show that the SINR values experienced by a typical user with respect to different base stations of a Poissonian cellular network are related to a specific instance of the so-called two-parameter Poisson-Dirichlet process. This process has many interesting properties as well as applications in various fields. We give examples of several results proved for this process that are of immediate or …

Prior Ordering and Monotonicity in Dirichlet Bandits

One of two independent stochastic processes (arms) are to be selected at each of n stages. The selection is sequential and depends on past observations as well as the prior information. Observations from arm i are independent given a distribution P_i, and, following Clayton and Berry (1985), P_i's have independent Dirichlet process priors. The objective is to maximize the expected future-discounted sum of the n observations. We study structural properties of the bandit, in particular how the maximum expected payoff and the optimal strategy vary with the Dirichlet process priors. The main resul…

A Bayesian View of the Poisson-Dirichlet Process

The two parameter Poisson-Dirichlet Process (PDP), a generalisation of the Dirichlet Process, is increasingly being used for probabilistic modelling in discrete areas such as language technology, bioinformatics, and image analysis. There is a rich literature about the PDP and its derivative distributions such as the Chinese Restaurant Process (CRP). This article reviews some of the basic theory and then the major results needed for Bayesian modelling of discrete problems including details of priors, posteriors and computation. The PDP allows one to build distributions over countable partitions…