Classifying lattice walks restricted to the quarter plane

This work considers the nature of generating functions of random lattice walks restricted to the first quadrant. In particular, we find combinatorial criteria to decide if related series are algebraic, transcendental holonomic or otherwise. Complete results for walks taking their steps in a maximum of three directions of restricted amplitude are given, as is a well-supported conjecture for all walks with steps taken from a subset of ²{0,±1}. New enumerative results are presented for several classes, each obtained with a variant of the kernel method.     

Analysis of exact tail asymptotics for singular random walks in the quarter plane

In this paper, we consider all singular cases of random walks in the quarter plane. Specifically, exact light tail asymptotics for stationary probabilities are obtained for all singular random walks.     

On certainp-harmonic functions in the plane

A representation theorem is given for certainp-harmonic functions (1<p<) in the plane, near a corner of the domain of definition. The proof uses stream functions and the hodograph transformation. The stream function of ap-harmonic function isp-harmonic, where 1/p+1/p=1. Solutions of the formr ^k f(ø) (polar coord.) are crucial for the representation.     

Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane

A random walk in $\mathbf{Z}_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Fig. 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j) \in \mathbf{Z}_+^2 $ at a given time $k\ge 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\rightarrow \infty $ . The equivalence of two typical ways of conditioning this random walk to never reach the axes is established. The results are also applied to the analysis of the voter model with two candidates and initially, in the population $\mathbf{Z}$ , four connected blocks of same opinions. Then, a citizen changes his mind at a rate proportional to the number of his neighbors that disagree with him. Namely, the passage from four to two blocks of opinions is studied.     

On functions with small differences

An Abel-Tauber theorem is proved and applied to multiplicative arithmetic functions.     

On one operator of convolution type in a quarter plane with an annihilatable symbol

We examine one class of Wiener-Hopf operators in a quarter plane, not subject to Noether's theory. We construct an unbounded inverse and describe the image of an operator from the class indicated.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 559–570, October, 1976.     

On the one-sided convergence of the Robbins-Monro process with small steps

In this paper, we consider the one-sided convergence of the Robbins-Monro process in the cases that have not been investigated earlier.     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

On a classification of meromorphic functions defined in the finite plane

We consider a subclass of meromorphic functions in the finite plane ¦z¦ < ∞, which generalizes Yosida's subclass of functions of the first category. We prove necessary and sufficient conditions for a function to belong to this subclass.     

On the error term for the counting functions of finite Abelian groups

Leta(n) denote the number of non-isomorphic Abelian groups withn elements, and Δ(x) (resp. Δ _x ) appropriate error terms in the asymptotic formulas for the counting function \(\sum\nolimits_{n \leqslant x} {a(n)} (resp. \sum\nolimits_{m n \leqslant x} {a(m)} a(n))\) . Sharp bounds for $$\int\limits_1^X {\Delta (x) dx} , \int\limits_1^X {\Delta _{ 1} (x) dx} ,\int\limits_1^X {\Delta _1^2 (x) dx} $$ are given by using results on power moments of the Riemann zeta-function.     

On the mixed boundary-value problem for harmonic functions in plane domains

This paper considers the problem of the determination of a harmonic function in a simply connected plane domain when the values of the function are known on some arcs of the boundary and the values of the normal derivative are known on the remaining boundary. We first present the solution in theoretical form and then show how to compute the solution with the aid of rapidly convergent series. The coefficients of these series are Fourier coefficients of certain functions and can be estimated by using the Fast Fourier Transform. The examples considered in the last part of the paper emphasize the advantages of the method presented in this paper for solving the mixed boundary-value problem as compared to other methods used for this purpose.

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Résumé On considère le problème suivant: trouver une fonction harmonique dans un domaine simplement connexeDR ² dont la frontièreS est assez régulière, en connaissant les valeurs de la fonction sur une partieS ₁ de la frontière et les valeurs de la dérivée normale sur la complémentaire de366-1. La première partie présente des résultats théoriques liés à ce probléme. La deuxième partie s'appuie sur une représentation de la solution à l'aide de certaines séries de fonctions rapidement convergentes. Les coefficients des séries utilisées sont même les coefficients de Fourier de certaines fonctions, et leur calcul peut être effectué en utilisant la transformation de Fourier rapide (Fast Fourier Transform). Les exemples considérés dans la dernière partie de l'article mettent en évidence les avantages de la méthode présentée par rapport à d'autres méthodes employées pour la résolution du problème mixte.