Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere

Let u be a map from a Riemann surface M to a Riemannian manifold N and $\alpha >1$, the α energy functional is defined as

$E_{\alpha }(u)=\frac{1}{2}\underset{M}{\int }[{(1+|▽u{|}^2)}^{\alpha }?1]dV.$

We call $u_{\alpha }$ a sequence of Sacks–Uhlenbeck maps if $u_{\alpha }$ are critical points of $E_{\alpha }$ and

$\underset{\alpha >1}{\sup }?E_{\alpha }(u_{\alpha })<\infty .$

In this paper, we show the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up, if the target N is a sphere $S^{K?1}$. The energy identity can be used to give an alternative proof of Perelman's result [15] that the Ricci flow from a compact orientable prime non-aspherical 3-dimensional manifold becomes extinct in finite time (cf. [3], [4]).     

Occupation times of the Ornstein–Uhlenbeck process: Functional PCA and evidence from electricity prices

We discuss the functional principal component analysis (FPCA) of the occupation times of the Ornstein–Uhlenbeck process. For the eigenvalue problem of the covariance operator of the occupation times we derive the corresponding integral equation in the large time limit and we solve numerically for the principal components. The formulation applies the path-integral approach of Feynman and Kac. The principal components are compared with those from empirical electricity price processes on energy markets. The results indicate that FPCA of the occupation times is a suitable tool in stochastic energy modeling to generate moderately-sized scenario trees.     

Generalized Ornstein–Uhlenbeck process by Doob’s theorem and the time evolution of financial prices

We generalize the Ornstein–Uhlenbeck (OU) process using Doob’s theorem. We relax the Gaussian and stationary conditions, assuming a linear and time-homogeneous process. The proposed generalization retains much of the simplicity of the original stochastic process, while exhibiting a somewhat richer behavior. Analytical results are obtained using transition probability and the characteristic function formalism and compared with empirical stock market data, which are notorious for the non-Gaussian behavior. The analysis focus on the decay patterns and the convergence study of the first four cumulants considering the logarithmic returns of stock prices. It is shown that the proposed model offers a good improvement over the classical OU model.     

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Abstract

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.     

Ornstein–Uhlenbeck process,Cauchy process,and Ornstein–Uhlenbeck–Cauchy process on a circle

By a simple mathematical method, we obtain the transition probability density functions of the Ornstein–Uhlenbeck process, Cauchy process, and Ornstein–Uhlenbeck–Cauchy process on a circle.     

Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

Let μ be an invariant measure for the transition semigroup (Pt) of the Markov family defined by the Ornstein-Uhlenbeck type equation

     

Sample path moderate deviations for the cumulative fluid produced by an increasing number of exponential on-off sources

We consider the superposition of the cumulative fluid generated by an increasing number of stationary iid on-off sources with exponential iid on- and off-time distributions. We establish a family of sample path large deviation principles when the fluid is centered and then scaled with a factor between the inverse of the number of sources and its square root. The common rate function in this family also appears in a large deviation principle for the tail probabilities of an integrated Ornstein–Uhlenbeck process. When the produced fluid is centered and scaled with the square root of the inverse of the number of sources it converges to this integrated Ornstein–Uhlenbeck process in distribution. We discuss several representations of the rate function. We apply the results to queueing systems loaded with on-off traffic and approaching critical loading.      

Characterization of Stochastic Bifurcations in a Simple Biological Oscillator

This study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first-passage-time problem of the Ornstein–Uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density. Stochastic phase-locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase. A firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel.