We extend results on time-rescaled occupation time fluctuation limits of the $(d,α, β)$-branching particle system $(0<α\leq 2, 0<β\leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>α/ β$ only, since the particle system becomes locally extinct if $d\le α/ β$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>α/β$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<α(1+β)/β$ and $d<α(2+β)/(1+β)$, respectively) the limits are determined by non-Lévy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\cal S}'(R^d)$-valued Lévy processes in the Lebesgue case, stable processes constant in time on $(0,\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $β=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes.
Shapley (1953) introduced two-player zero-sum discounted stochastic games, henceforth stochastic games, a model where a state variable follows a two-controlled Markov chain, the players receive rewards at each stage which add up to $0$, and each maximizes the normalized $\la$-discounted sum of stage rewards, for some fixed discount rate $\la\in(0,1]$. In this paper, we study asymptotic occupation measures arising in these games, as the discount rate goes to $0$.
The L2-approximation of occupation and local times of a symmetric $α$-stable L{é}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < $α$ $\le$ 1, but only rate optimal for 1 < $α$ $\le$ 2. For this, the exact convergence of the L2-approximation error is proven with explicit constants.
We shall establish a large deviation principle for some occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. As a consequence, we obtain the exact rate of exponential ergodicity of the stochastic system under $τ$-topology.
We establish a large deviation principle for the occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. The proof is based on a hyper-exponential recurrence criterion. Our result indicates a phenomenon that strong dissipation beats heavy tailed noises to produce a large deviation, it seems to us that this phenomenon has not been reported in the known literatures.
A class of Fleming-Viot processes with decaying sampling rates and $α$-stable motions that correspond to distributions with growing populations are introduced and analyzed. Almost sure long-time scaling limits for these processes are developed, addressing the question of long-time population distribution for growing populations. Asymptotics in higher orders are investigated. Convergence of particle location occupation and inhabitation time processes are also addressed and related by way of the historical process. The basic results and techniques allow general Feller motion/mutation and may app…
We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the pre…
Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $α$-stable noises.
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-value…
Stochastic processes time-changed by an inverse subordinator have been suggested as a way to model the price of assets in illiquid markets, where the jumps of the subordinator correspond to periods of time where one is unable to sell an asset. We develop an excursion theory for time-changed reflected Brownian motion and use this to express the price of certain European options with Parisian barrier condition in terms of solutions of a time-fractional PDE. We provide a general description of the occupation measures of time-changed processes and use this to prove a Ray-Knight theorem for the occ…
We consider a particle system with weights and the scaling limits derived from its occupation time. We let the particles perform independent recurrent Lévy motions and we assume that their initial positions and weights are given by a Poisson point process. In the limit we obtain a number of recently discovered stationary stable self-similar processes recently studied in [SAM1] and [SAM2]. We also observe very different limit processes depending on whether the function, whose occupation time is considered, integrates to zero or not.
Occupation-Time Fluctuations of an Age-Dependent Branching System Driven by Poisson Immigration
Centro de Investigación en Matemáticas（数学研究中心）；Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”（索非亚“圣克利门特·奥赫里德斯基”大学数学与信息学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与情报研究所）
Gamma Tilting Calculus for GGC and Dirichlet means with applications to Linnik processes and Occupation Time Laws for Randomly Skewed Bessel Processes and Bridges
本文发展了GGC和Dirichlet过程平均泛函的一些一般演算。然后，它继续通过调查的积极Linnik随机变量，更一般的随机变量来自一个稳定的从属GGC从属成分，建立各种分布之间的等价关系，这些模型和现象连接到本地时间和占领时间的什么被定义为随机偏斜贝塞尔过程和桥梁。这产生了大量有趣的身份和明确的密度公式，这些模型。随机偏斜贝塞尔过程和桥可以被看作是Barlow，Pitman and Yor（1989）和Pitman and Yor（1997）中的p偏斜过程和桥的随机化，并且可以通过指数倾斜自然产生。作为一个特殊结果，证明了p-斜随机Bessel过程或（广义）桥的占有时间与非平凡随机斜过程的占有时间在分布上是等价的.
We investigate a family of multiple-stable processes that may exhibit either long-range or short-range dependence, depending on the parameters. There are two parameters for the processes, the memory parameter $β\in(0,1)$ and the multiplicity parameter $p\in\mathbb N$. We investigate the macroscopic limit of extremes of the process, in terms of convergence of random sup-measures, for the full range of parameters. Our results show that (i) the extremes of the process exhibit long-range dependence when $β_p := pβ-p+1\in(0,1)$, with a new family of random sup-measures arising in the limit, (ii) th…
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
