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Register a new userStable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II
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This paper is a continuation of our recent paper (Elect. J. Probab. 24 (2019), no. 141) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes $(X_t)_{tgeq 0}$ with branching mechanisms of infinite second moment. In the aforementioned paper, we proved stable central limit theorems for $X_t(f) $ for some functions $f$ of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth. In this note, we show that the limit stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth.
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In this paper, we study the asymptotic behavior of a supercritical $(xi,psi)$-superprocess $(X_t)_{tgeq 0}$ whose underlying spatial motion $xi$ is an Ornstein-Uhlenbeck process on $mathbb R^d$ with generator $L = frac{1}{2}sigma^2Delta - b x cdot abla$ where $sigma, b >0$; and whose branching mechanism $psi$ satisfies Greys condition and some perturbation condition which guarantees that, when $zto 0$, $psi(z)=-alpha z + eta z^{1+beta} (1+o(1))$ with $alpha > 0$, $eta>0$ and $betain (0, 1)$. Some law of large numbers and $(1+beta)$-stable central limit theorems are established for $(X_t(f) )_{tgeq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.
The Ornstein-Uhlenbeck process can be seen as a paradigm of a finite-variance and statistically stationary rough random walk. Furthermore, it is defined as the unique solution of a Markovian stochastic dynamics and shares the same local regularity as the one of the Brownian motion. Based on previous works, we propose to include in the framework of one of its generalization, the so-called fractional Ornstein-Uhlenbeck process, some Multifractal corrections, using a Gaussian Multiplicative Chaos. The aforementioned process, called a Multifractal fractional Ornstein-Uhlenbeck process, is a statistically stationary finite-variance process. Its underlying dynamics is non-Markovian, although non-anticipating and causal. The numerical scheme and theoretical approach are based on a regularization procedure, that gives a meaning to this dynamical evolution, which unique solution converges towards a well-behaved stochastic process.
The paper is concerned with the properties of solutions to linear evolution equation perturbed by cylindrical Levy processes. It turns out that solutions, under rather weak requirements, do not have c`adl`ag modification. Some natural open questions are also stated.
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied.
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