Stationary processes whose filtrations are standard

Explicit construction of a dynamic Bessel bridge of dimension 3

Given a deterministically time-changed Brownian motion $Z$ starting from 1, whose time-change $V(t)$ satisfies $V(t) > t$ for all $t > 0$, we perform an explicit construction of a process $X$ which is Brownian motion in its own filtration and that hits zero for the first time at $V(τ)$, where $τ:= \inf\{t>0: Z_t =0\}$. We also provide the semimartingale decomposition of $X$ under the filtration jointly generated by $X$ and $Z$. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process $X$ may be viewed as the analogue of a 3-dimension…

与莱维过程相关的桥型过程及其分解

Bridge-Type Processes Associated with Lévy Processes and Their Decompositions

研究一类终端固定值为随机变量的桥型过程，分析其在自然滤波下的半鞅分解，探讨非高斯莱维驱动下的桥型过程特性。

Partially Gaussian Stationary Stochastic Processes in Discrete Time

We present here an elementary example, for every fixed positive integer $k,$ of a strictly stationary nongaussian stochastic process in discrete time, all of whose $k$-marginals are gaussian.

Product-form stationary distributions for deficiency zero networks with non-mass action kinetics

In many applications, for example when computing statistics of fast subsystems in a multiscale setting, we wish to find the stationary distributions of systems of continuous time Markov chains. Here we present a class of models that appears naturally in certain averaging approaches whose stationary distributions can be computed explicitly. In particular, we study continuous time Markov chain models for biochemical interaction systems with non-mass action kinetics whose network satisfies a certain constraint. Analogous with previous related results, the distributions can be written in product f…

Quasi-stationary behavior for an hybrid model of chemostat: the Crump-Young model

The Crump-Young model consists of two fully coupled stochastic processes modeling the substrate and microorganisms dynamics in a chemostat. Substrate evolves following an ordinary differential equation whose coefficients depend of microorganisms number. Microorganisms are modeled though a pure jump process whose the jump rates depend on the substrate concentration. It goes to extinction almost-surely in the sense that microorganism population vanishes. In this work, we show that, conditionally on the non-extinction, its distribution converges exponentially fast to a quasi-stationary distributi…

Ergodic Theory on Stationary Random Graphs

A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic function. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.

A Primer on Homogenization of Elliptic PDEs with Stationary and Ergodic Random Coefficient Functions

We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.

随机游走的 $p$-进标度极限中由几何诱导的临界性

Geometry-induced criticality in $p$-adic scaling limits of random walks

本文研究 $\mathbb{Q}_p^2$ 上分层随机游走标度极限中由各向异性几何导致的临界现象，揭示双尺度滤过坍缩时扩散常数跳跃的结构机制。

Exponential concentration for zeroes of stationary Gaussian processes

We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $ηT$ of its mean value, up to an exponentially small in $T$ probability.