Independent Ornstein-Uhlenbeck Research Articles

Stability of Regularized Hastings–Levitov Aggregation in the Subcritical Regime

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE(α,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha },\\eta )$$\\end{document} of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha })$$\\end{document} and continuum versions of the family of dielectric-breakdown models, where the local attachment intensity for new particles is specified as a negative power -η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\eta $$\\end{document} of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner–Kufarev equation, where the driving measure is made to depend on the solution and on the parameter ζ=α+η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta }={\\alpha }+\\eta $$\\end{document}. Our results are subject to a subcriticality condition ζ⩽1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta }\\leqslant 1$$\\end{document}: this includes HL(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha })$$\\end{document} for α⩽1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\alpha }\\leqslant 1$$\\end{document} and also the case α=2,η=-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\alpha }=2,\\eta =-1$$\\end{document} corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\alpha })$$\\end{document} at α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\alpha }=1$$\\end{document}, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein–Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if ζ⩽1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta }\\leqslant 1$$\\end{document}.

Asymptotics of Yule’s nonsense correlation for Ornstein-Uhlenbeck paths: A Wiener chaos approach

In this paper, we study the distribution of the so-called Yule's nonsense correlation on a time interval $[0,T]$ for a time horizon $T>0$ , when $T$ is large, for a pair $(X_{1},X_{2})$ of independent Ornstein-Uhlenbeck processes. This statistic is by definition equal to : \begin{equation*} \rho (T):=\frac{Y_{12}(T)}{\sqrt{Y_{11}(T)}\sqrt{Y_{22}(T)}}, \end{equation*} where the random variables $Y_{ij}(T)$, $i,j=1,2$ are defined as \begin{equation*} Y_{ij}(T):=\int_{0}^{T}X_{i}(u)X_{j}(u)du-T\bar{X}_{i}\bar{X_{j}}, \bar{X}_{i}:=\frac{1}{T}\int_{0}^{T}X_{i}(u)du. \end{equation*} We assume $X_{1}$ and $X_{2}$ have the same drift parameter $\theta >0$. We also study the asymptotic law of a discrete-type version of $\rho (T)$, where $Y_{ij}(T)$ above are replaced by their Riemann-sum discretizations. In this case, conditions are provided for how the discretization (in-fill) step relates to the long horizon $T$. We establish identical normal asymptotics for standardized $\rho (T)$ and its discrete-data version. The asymptotic variance of $\rho (T)T^{1/2}$ is $\theta ^{-1}$. We also establish speeds of convergence in the Kolmogorov distance, which are of Berry-Esseen-type (constant*$T^{-1/2}$) except for a $\ln T$ factor. Our method is to use the properties of Wiener-chaos variables, since $\rho (T)$ and its discrete version are comprised of ratios involving three such variables in the 2nd Wiener chaos. This methodology accesses the Kolmogorov distance thanks to a relation which stems from the connection between the Malliavin calculus and Stein's method on Wiener space.

Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC

We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the proposed scaling law to predict the performance of (terminated) SC-LDPC code ensembles under the more practical sliding window decoding. Finally, we extend this framework to predict the bit error rate (BER) and block error rate (BLER) of SC-LDPC code ensembles. The proposed scaling law yields very accurate predictions of the FER, BLER, and BER for both full BP and window decoding.

The genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selection

We study the genealogy of a solvable population model with $N$ particles on the real line which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around $a$ times its current position, where $a>0$ is a parameter of the model. Then, the $N$ rightmost new-born children are selected to form the next generation. We show that the genealogical trees of the process converge to those of a Beta coalescent as $N \to \infty$. The process we consider can be seen as a toy-model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein-Uhlenbeck processes. The parameter $a$ is akin to the pulling strength of the Ornstein-Uhlenbeck motion.

On the Lamperti Transform of the Fractional Brownian Sheet

Abstract In 1962 Lamperti introduced a transformation that associates to every non-trivial self-similar process a strictly stationary process. This transform has been widely studied for Gaussian processes and in particular for fractional Brownian motion. Our aim is to analyze various properties of the Lamperti transform of the fractional Brownian sheet. We give the stochastic differential equation satisfied by this transform and we represent it as a series of independent Ornstein-Uhlenbeck sheets.

Stochastic Differential Equation of Earthquakes Series

This work is devoted to modeling earthquake time series. We propose a stochastic differential equation based on the superposition of independent Ornstein–Uhlenbeck processes driven by a $$\Gamma (\alpha, \beta )$$ process. Superposition of independent $$\Gamma (\alpha, \beta )$$ Ornstein–Uhlenbeck processes offer analytic flexibility and provides a class of continuous time processes capable of exhibiting long memory behavior. The stochastic differential equation is applied to the study of earthquakes by fitting the superposed $$\Gamma (\alpha, \beta )$$ Ornstein–Uhlenbeck model to earthquake sequences in South America containing very large events (Mw $$\ge $$ 8). We obtained very good fit of the observed magnitudes of the earthquakes with the stochastic differential equations, which supports the use of this methodology for the study of earthquakes sequence.

Stochastic differential equations applied to the study of geophysical and financial time series

This work is devoted to the study of modeling geophysical and financial time series. We propose a stochastic differential equation arising from the superposition of independent Ornstein–Uhlenbeck processes driven by a Γ(a,b) process. Superposition of independent Γ(a,b) Ornstein–Uhlenbeck processes offers analytic flexibility and provides a class of continuous time processes capable of exhibiting long memory behavior. The stochastic differential equation is applied to geophysics and finance by fitting the superposed Γ(a,b) Ornstein–Uhlenbeck model to typical geophysical and financial time series.

Consecutive minors for Dyson’s Brownian motions

In 1962, Dyson (1962) introduced dynamics in random matrix models, in particular into GUE (also for β=1 and 4), by letting the entries evolve according to independent Ornstein–Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: β=1,2,4. It is also shown that the spectra of three consecutive minors ceases to be Markovian for β=2,4.

The Local Time of the Fractional Ornstein-Uhlenbeck Process

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Non-Markovian Effect on the Classical and Quantum Correlations

We evaluate the dynamics of quantum and classical correlations in the presence of non-Markovian noises. By considering an entangled pair of spins in the noise environment described by Ornstein—Uhlenbeck processes, we show that the quantum discord of some states is completely unaffected by independent Ornstein—Uhlenbeck noises for long intervals of time, and that the inevitable onset of the sudden decrease of the quantum discord can be substantially delayed by the decrease of the noise bandwidth γ, where γ−1 = τc defines the environment's finite correlation time of the noise.

An equivalent representation of the Brown–Resnick process

Brown and Resnick (1977) introduce a max-stable process that is obtained as a limit of maxima of independent Ornstein–Uhlenbeck processes. As shown in Kabluchko et al. (2009) this process is dissipative and it therefore admits a mixed moving maxima representation. We show that the distribution of the spectral functions in this representation equals a well-known diffusion, namely a standard Brownian motion with drift conditional on taking negative values only. This can be used for fast simulation methods.

Tests of Non Linear Gaussian Term Structure Models

After the 2007-2009 financial crisis Government bond yields in US, Germany, UK and elsewhere have approached Japan-type Government bond yields, thus challenging known dynamic term structure models that cannot constrain yields to be always positive. Therefore this paper tests term structure models whereby the instantaneous short interest rate is a non-negative, non-linear function of latent factors that follow independent Ornstein-Uhlenbeck processes. These models, which we refer to as non-affine Gaussian term structure models (NAGTSM) encompass the quadratic, Black and Black-Karasinski models. The empirical evidence reveals that three factor NAGTSM and Vasicek models fi t US and German yields to a similar degree, except for the very low short maturity US yields after 2007, which the Vasicek model cannot fit very well. Three factor quadratic models generally display the worst fit to US and German yields.

Perturbation analysis of an M/M/1 queue in a diffusion random environment

We study in this paper an M/M/1 queue whose server rate depends upon the state of an independent Ornstein–Uhlenbeck diffusion process (X(t)) so that its value at time t is μ φ(X(t)), where φ(x) is some bounded function and μ>0. We first establish the differential system for the conditional probability density functions of the couple (L(t),X(t)) in the stationary regime, where L(t) is the number of customers in the system at time t. By assuming that φ(x) is defined by φ(x)=1−ε((x ∧ a/ε)∨(−b/ε)) for some positive real numbers a, b and ε, we show that the above differential system has a unique solution under some condition on a and b. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when φ is replaced with Φ(x)=1−ε x for sufficiently small ε. We finally perform a perturbation analysis of this latter solution for small ε. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to $\mu(1-\varepsilon {\mathbb{E}}(X(t)))$ .

Transparent boundary conditions as dissipative subgrid closures for the spectral representation of scalar advection by shear flows

We consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space. When the velocity field has small support, the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves. We close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves. This closure is studied numerically in terms of energy dissipation rate and energy spectrum, both for a time-independent velocity field and for a time-dependent synthetic velocity field whose Fourier coefficients follow independent Ornstein-Uhlenbeck stochastic processes.

Stochastic Models for the Kinematics of Moisture Transport and Condensation in Homogeneous Turbulent Flows

Abstract The transport of a condensing passive scalar is studied as a prototype model for the kinematics of moisture transport on isentropic surfaces. Condensation occurs whenever the scalar concentration exceeds a specified local saturation value. Since condensation rates are strongly nonlinear functions of moisture content, the mean moisture flux is generally not diffusive. To relate the mean moisture content, mean condensation rate, and mean moisture flux to statistics of the advecting velocity field, a one-dimensional stochastic model is developed in which the Lagrangian velocities of air parcels are independent Ornstein–Uhlenbeck (Gaussian colored noise) processes. The mean moisture evolution equation for the stochastic model is derived in the Brownian and ballistic limits of small and large Lagrangian velocity correlation time. The evolution equation involves expressions for the mean moisture flux and mean condensation rate that are nonlocal but remarkably simple. In a series of simulations of homogeneous two-dimensional turbulence, the dependence of mean moisture flux and mean condensation rate on mean saturation deficit is shown to be reproducible by the one-dimensional stochastic model, provided eddy length and time scales are taken as given. For nonzero Lagrangian velocity correlation times, condensation reduces the mean moisture flux for a given mean moisture gradient compared with the mean flux of a noncondensing scalar.

SOME FUNCTIONAL LIMIT THEOREMS FOR THE INFINITE SERIES OF OU PROCESSES

This paper obtains functional modulus of continuity and Strassen's functional LIL of the infinite series of independent Ornstein-Uhlenbeck processes, which also imply the Levy's exact modulus of continuity and LIL of this process respectively.

Almost sure continuity of infinite series of independent two-parameter Ornstein-Uhlenbeck processes

This study on the conditions for the convergence and almost-sure continuity of infinite series of independent two-parameter Ornstein-Uhlenbeck processes yielded sufficient conditions similar to that for independent one-parameter Ornstein-Uhlenbeck processes.

Almost sure continuity of infinite series of independent two-parameter Ornstein-Uhlenbeck processes

Almost sure continuity of infinite series of independent two-parameter Ornstein-Uhlenbeck processes

Singular-values of matrix-valued Ornstein–Uhlenbeck processes

The system of singular-values of a square matrix whose components are independent Ornstein–Uhlenbeck processes corresponds to a diffusion model of interacting particles. We show that the weak limit, as the dimension of the matrix tends to infinity, of the associated empirical measure process is a deterministic measure-valued process and converges to a fixed law as the time t tends to infinity.

On the fractal nature of increments of the infinite series of ou processes related to the Chung LIL

The author proves that the set of points where the Chung type LIL fails for the path of the infinite series of independent Ornstein-Uhlenbeck processes is a random fractal, and eval-uates its Hausdorff dimension.