Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Asymptotics of the zeros of degenerate hypergeometric functions

We find the asymptotics of the zeros of the degenerate hypergeometric function (the Kummer function) Φ(a, c; z) and indicate a method for numbering all of its zeros consistent with the asymptotics. This is done for the whole class of parameters a and c such that the set of zeros is infinite. As a corollary, we obtain the class of sine-type functions with unfamiliar asymptotics of their zeros. Also we prove a number of nonasymptotic properties of the zeros of the function Φ.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

A simple global model for the general circulation of the atmosphere

In this article we present a new method of Rossby asymptotics for the equations of the atmosphere similar to the geostrophic asymptotics. We depart from the classical geostrophics (see J. G. Charney [5] and our previous article [29]) by considering an asymptotics valid for the whole atmosphere, not only in midlatitude regions, and by taking into account the spherical form of the earth. We obtain in this way a very simple global circulation model of the atmosphere for which the equations of motion for wind and temperature are linear evolution equations similar to the linear Stokes equations. Furthermore, the solutions are independent of longitude, and winds travel exactly to the east or to the west. In this mathematically oriented article, we do not discuss the physical significance of the model that we derive except for observing that this picture coincides in general terms with the classically averaged data obtained by experimental measurements. We also note that different global geostrophic asymptotics (called planetary geostrophic asymptotics) are considered elsewhere in the literature, for example, in J. Pedlosky [33]. In a less mathematically rigorous way, H. Lamb [21] observed that the only globally valid geostrophic flow that maintains a slow time scale is zonally symmetric (see the comments in N. Phillips [35] and an explicit derivation in H. Jeffreys [19]). © 1997 John Wiley & Sons, Inc.     

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

We study a system of discrete Painlevé V equations via the Riemann–Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann–Hilbert problem formulation. The asymptotics of the discrete Riemann–Hilbert problem is derived via the nonlinear steepest descent method of Deift and Zhou. In the analysis, a parametrix is constructed in terms of specific Painlevé V transcendents. As a result, the asymptotics of the dPV transcendents are represented in terms of the PV transcendents. In the special case, our result confirms a conjecture of Borodin, that the difference Schlesinger equations converge to the differential Schlesinger equations at the solution level.     

Asymptotic summation of perturbed linear difference systems in critical case

We propose an asymptotic summation method for certain class of linear difference systems. Both the ideas of the centre manifold theory and the averaging method are used to construct the asymptotics for solutions. We illustrate the asymptotic summation method by constructing the asymptotics for solutions of certain higher order scalar difference equation with an oscillatory decreasing coefficient.     

Small deviations of iterated processes in the space of trajectories

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.     

Scattering by a Cylindrical Trap in the Critical Case

We study a two-dimensional analog of the Helmholtz resonator with walls of finite thickness in the critical case, for which there exists a frequency which is simultaneously the limit of poles generated both by the bounded component of the resonator and by a narrow communication channel. Under the assumption that the limit frequency is a simple frequency for the bounded component, by using the method of matched asymptotic expansions, we construct asymptotics for the two sequences of poles converging to this frequency. We obtain explicit formulas for the leading terms of the asymptotics of poles and for the solution of the scattering problem.     

On asymptotic behavior of probabilities of large and moderate deviations for some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of large deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables. When these conditions do not hold, the asymptotics of large deviations for iterated processes are quite different. When the iterated process is a homogeneous process with independent increments in which time is replaced by random one, the behavior of large and moderate deviations is studied in the case of finite variance. For this case, the following one-sided moment restriction are considered: the Cramèr condition, the Linnik condition, and the existence of moment of order p > 2 for a positive part. Bibliography: 6 titles.     

Moving fronts in integro-parabolic reaction-advection-diffusion equations

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.     

Calculation and estimation of the Poisson kernel

We provide a simple method for obtaining boundary asymptotics of the Poisson kernel on a domain in RN.     

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L² convergence of Szegő asymptotics on the spectrum.     

Extremal Polynomials on Discrete Sets

We study asymptotics for orthogonal polynomials and other extremalpolynomials on infinite discrete sets, typical examples beingthe Meixner polynomials and the Charlier polynomials. Followingideas of Rakhmanov, Dragnev and Saff, weshow that the asymptoticbehaviour is governed by a constrained extremal energy problemfor logarithmic potentials, which can be solved explicitly.We give formulas for the contracted zero distributions, thenth root asymptotics and the asymptotics of the largest zeros.1991 Mathematics Subject Classification: 42C05, 33C25, 31A15.     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation

Traveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading-order behavior. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading-order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading-order behavior. We calculate the oscillation behavior for Toda woodpile chains, and compare the results to exponential asymptotics based on previous methods from the literature: long-wave approximation and tanh-fitting. We then use numerical analytic continuation methods based on Padé approximants and the adaptive Antoulas–Anderson (AAA) method. These methods are shown to produce accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. Exponential asymptotics using an AAA approximation for the leading-order behavior is then applied to study granular woodpile chains, including chains with Hertzian interactions—this method is able to calculate behavior that could not be accurately approximated in previous studies.     

The Dixmier trace and asymptotics of zeta functions

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p>1 that the asymptotics of the zeta function determines an ideal strictly larger than Lp,∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.     

Exact small ball asymptotics in weighted <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-norm for some Gaussian processes

We find the exact small ball asymptotics under weighted L ₂-norm for a wide class of Gaussian processes which generate boundary-value problems for ordinary differential equations. Sharp constants in the asymptotics are derived for a number of processes connected with special functions. Bibliography: 23 titles.     

Precise asymptotics of weighted sequences and their applications

We study an asymptotic property of weighted sequences of nonnegative functions which extends and unifies previous results concerned with precise asymptotics. As applications, we prove precise asymptotics for partial sums of independent identically distributed classical, noncommutative and free random variables.

     

Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann‐Hilbert Problem Approach

We study the asymptotics of singular values and singular functions of a finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann‐Hilbert problem (RHP) and the steepest‐descent method of Deift‐Zhou. We obtain a family of matrix RHPs depending on the spectral parameter λ and show that the singular values of the FHT coincide with the values of λ for which the RHP is not solvable. Expressing the leading‐order solution as λ → 0 of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading‐order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results. © 2016 Wiley Periodicals, Inc.     

Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.