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.     

How to test that a given process is an Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - We show asymptotic distributions of the residual process in Ornstein–Uhlenbeck model, when the model is true. These distributions are of...     

Fourier series and the δ process

We discuss the effect of a particular sequence acceleration method, the δ²${\delta }^2$ process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points.     

The continuous-time triangular Pólya process

We study poissonized triangular (reducible) urns on two colors, which we take to be white and blue. We analyze the number of white and blue balls after a certain period of time has elapsed. We show that for balanced processes in this class, a different scaling is needed for each color to produce nontrivial limits, contrary to the distributions in the usual irreducible urns which only require the same scaling for both colors. The limit distributions (of the scaled variables) underlying triangular urns are Gamma. The technique we use couples partial differential equations with the method of moments applied in a bootstrapped manner to produce exact and asymptotic moments. For the dominant color, we get exact moments, while relaxing the balance condition. The exact moments include alternating signs and Stirling numbers of the second kind.     

The Cayley–Dickson process for dialgebras

We adapt the algorithm of Kolesnikov and Pozhidaev, which converts a polynomial identity for algebras into the corresponding identities for dialgebras, to the Cayley–Dickson doubling process. We obtain a generalization of this process to the setting of dialgebras, establish some of its basic properties, and construct dialgebra analogues of the quaternions and octonions.     

Invariant measures for the strong-facilitated exclusion process

Consider a generalized model of the facilitated exclusion process, which is a onedimensional exclusion process with a dynamical constraint that prevents the particle at site x from jumping to x+1 (or x-1) if the sites x-1, x-2 (or x+1, x+2) are empty. It is nongradient and lacks invariant measures of product form. The purpose of this paper is to identify the invariant measures and to show that they satisfy both exponential decay of correlations and equivalence of ensembles. These properties will...     

What is a multi-parameter renewal process?

The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.     

A functional LIL for integrated α stable process

Let {X（t）,t ∈ R＋} be an integrated α stable process. In this paper, a functional law of the iterated logarithm （LIL） is derived via estimating the small ball probability of X. As a corollary,, the classical Chung LIL of X is obtained. Furthermore, some results about the weighted occupation measure of X（t） are established.     

Krylov–Safonov estimates for a degenerate diffusion process

This paper proves a Krylov–Safonov estimate for a multidimensional diffusion process whose diffusion coefficients are degenerate on the boundary. As applications the existence and uniqueness of invariant probability measures for the process and Hölder estimates for the associated partial differential equation are obtained.     

Vonnegut's spraying fountain—an oxygen-pressure dependent chemical process

Zusammenfassung Vonneguts klassisches Experiment über das Zerstäuben kleinster positiv hoch geladener Wassertröpfchen aus einer Kapillare wurde in einem geschlossenen Gefäss unter kontrollierten Bedingungen wiederholt. Die Zerstäubungsversuche wurden bei verschiedenen Mischungsverhältnissen des die Kapillare umgebenden Stickstoff-Sauerstoff-Gemisches ausgeführt, wobei Nebelproben gesammelt und analysiert wurden. Die erhaltenen Resultate deuten auf eine mögliche Reaktion des an der Kapillarenspitze in der Corona aktivierten Stickstoffes mit Wasser. Dies führt zur Bildung von Nitrit- und Nitrationen im gesammelten Nebel. In Gegenwart kleiner Sauerstoffmengen wird das Nitrit quantitativ zu Nitrat oxydiert. Dieses für die Atmosphärenchemie möglicherweise bedeutsame Phänomen ist begleicet von einer anderen Erscheinung, die vielleicht in Beziehung steht mit der Bildung natürlicher Nebel. Es wird gezeigt, dass bei schrittweiser Senkung des Sauerstoff-Stickstoff-Druckes in der Umgebung der Kapillare die Flüssigkeit nicht mehr zerstäubt wird, sondern als Strahl auftritt. Der den Übergang kennzeichnende Druck hängt direkt von der Zusammensetzung des Gasgemisches ab. Im reinen Stickstoff findet unterhalb 760 torr bei den gewählten experimentellen Bedingungen kein Zerstäuben statt, wogegen in einer Atmosphäre, deren Zusammensetzung der Luft entspricht, der Übergang bei etwa 615 torr liegt, also bei einem Druck der 1700 m Meereshöhe entspricht.     

The density evolution of the killed McKean–Vlasov process

ABSTRACT

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.     

Discretization error for a two-sided reflected Lévy process

An obvious way to simulate a Lévy process X is to sample its increments over time 1 / n, thus constructing an approximating random walk \(X^{(n)}\). This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resulting process Y and regulators L, U at the lower and upper barriers at some fixed time. Under the weak assumption that \(X_\varepsilon /a_\varepsilon \) has a non-trivial weak limit for some scaling function \(a_\varepsilon \) as \(\varepsilon \downarrow 0\), it is proved in particular that \((Y_1-Y^{(n)}_n)/a_{1/n}\) converges weakly to \(\pm \, V\), where the sign depends on the last barrier visited. Here the limit V is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (Ann Appl Probab, 2018). Some further insight in the distribution of V is provided both theoretically and numerically.     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

Kohnen’s limit process for real-analytic Siegel modular forms

Kohnen introduced a limit process for Siegel modular forms that produces Jacobi forms. He asked if there is a space of real-analytic Siegel modular forms such that skew-holomorphic Jacobi forms arise via this limit process. In this paper, we initiate the study of harmonic skew-Maass–Jacobi forms and harmonic Siegel–Maass forms. We improve a result of Maass on the Fourier coefficients of harmonic Siegel–Maass forms, which allows us to establish a connection to harmonic skew-Maass–Jacobi forms. In particular, we answer Kohnen’s question in the affirmative.     

Detection of cellular aging in a Galton–Watson process

We consider the bifurcating Markov chain model introduced by Guyon to detect cellular aging from cell lineage. To take into account the possibility for a cell to die, we use an underlying super-critical binary Galton–Watson process to describe the evolution of the cell lineage. We give in this more general framework a weak law of large number, an invariance principle and thus fluctuation results for the average over all individuals in a given generation, or up to a given generation. We also prove that the fluctuations over each generation are independent. Then we present the natural modifications of the tests given by Guyon in cellular aging detection within the particular case of the auto-regressive model.     

The Pólya sum process: a Cox representation

In [1], Zessin constructed the so-called Pólya sum process via partial integration. Here we use the technique of integration by parts to the Pólya sum process to derive representations of the Pólya sum process as an infinitely divisible point process and a Cox process directed by an infinitely divisible random measure. This result is related to the question of the infinite divisibilty of a Cox process and the infinite divisibility of its directing measure. Finally we consider a scaling limit of the Pólya sum process and show that the limit satisfies an integration by parts formula, which we use to determine basic properties of this limit.     

A conditioned ornstein–uhlenbeck process on a hilbert space

An infinite system of stochastic differential equations is given, each of which defines a pinned Ornstein–Uhlenbeck process on R , and it is shown that the system defines a pinned Ornstein–Uhlenbeck process on a certain Hubert space     

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.