Narn-Rueih Shieh

Multifractality of products of geometric Ornstein-Uhlenbeck type processes

We investigate the properties of multifractal products of geometric Ornstein-Uhlenbeck (OU) processes driven by Lévy motion. The conditions on the mean, variance, and covariance functions of the resulting cumulative processes are interpreted in terms of the moment generating functions. We consider five cases of infinitely divisible distributions for the background driving Lévy processes, namely, the gamma and variance gamma distributions, the inverse Gaussian and normal inverse Gaussian distributions, and the z-distributions. We establish the corresponding scenarios for the limiting processes, including their Rényi functions and dependence structure.

Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times

By using a wavelet method we prove that the harmonisable-type N -parameter multifractional Brownian motion (mfBm) is a locally nondeterministic Gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (N, d)-mfBm and to obtain some new results concerning its sample path behavior.

Logarithmic multifractal spectrum of stable occupation measure

For a stable subordinator Yt of index [alpha], 0 1, and B[theta] [not equal to] [empty set][combining character] for 0[less-than-or-equals, slant][theta][less-than-or-equals, slant]1; moreover, dim B[theta]=Dim B[theta]=[alpha](1-[theta]1/(1-[alpha])).

Multifractal products of stationary diffusion processes

Abstract We investigate the properties of multifractal products of the exponential of stationary diffusion processes defined by stochastic differential equations with linear drift and certain form of the diffusion coefficient corresponding to a variety of marginal distributions. The conditions on the mean, variance and covariance functions of these processes are interpreted in terms of the moment generating functions. We provide three illustrative examples of normal, gamma and beta distributions. We establish the corresponding lognormal, log-gamma and log-beta scenarios for the limiting processes, including their Rényi functions and dependence structure.

Multifractal scaling of products of birth-death processes

We investigate the scaling properties of products of the exponential of birth–death processes with certain given marginal discrete distributions and covariance structures. The conditions on the mean, variance and covariance functions of the resulting cumulative processes are interpreted in terms of the moment generating functions. We provide four illustrative examples of Poisson, Pascal, binomial and hypergeometric distributions. We establish the corresponding log-Poisson, log-Pascal, log-binomial and log-hypergeometric scenarios for the limiting processes, including their Renyi functions and dependence properties.

RÉNYI FUNCTION FOR MULTIFRACTAL RANDOM FIELDS

This paper presents the basic scheme and the log-normal, log-gamma and log-negative-inverted-gamma scenarios to establish the Renyi function for infinite products of homogeneous isotropic random fields on Rn; in particular for random fields on the sphere in R3. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

Simulation of multifractal products of Ornstein-Uhlenbeck type processes

This paper investigates and provides evidence of the multifractal properties of products of the exponential of Ornstein–Uhlenbeck processes driven by Levy motion. We demonstrate in detail the construction of a multifractal process with gamma subordinator as the background driving Levy process. Simulations are performed for the scenarios corresponding to the normal inverse Gaussian, gamma and inverse Gaussian distributions. The log periodograms and Renyi functions of the simulated processes are also computed to investigate their multifractality.

Scaling limits for time-fractional diffusion-wave systems with random initial data

Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

Multifractal spectra of certain random Gibbs measures

We consider a random Gibbs measure [mu](d[eta],[omega]) generated by a certain sequence of random functions gn([eta],[omega]) on the configuration space of one-dimensional system of lattice particles. Under concrete conditions, we prove that, for almost sure [omega], [mu](d[eta],[omega]) has a non-random non-trivial multifractal spectrum. The basic idea is to relate our situation to random matrix products discussed in Ruelle (1979, Adv. Math. 32, 68-80).

Multifractal scenarios for products of geometric Ornstein-Uhlenbeck type processes

We investigate the properties of multifractal products of the exponential of Ornstein-Uhlenbeck processes driven by Levy motion. The conditions on the mean, variance and covariance functions of these processes are interpreted in terms of the moment generating functions. We provide two examples of tempered stable and normal tempered stable distributions. We establish the corresponding log-tempered stable and log-normal tempered stable scenarios, including their Renyi functions and dependence structures.