Some results on Gaussian Besov–Lipschitz spaces and Gaussian Triebel–Lizorkin spaces

In this paper we define Besov–Lipschitz and Triebel–Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein–Uhlenbeck and the Poisson–Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.     

An asymptotic functional-integral solution for the Schrödinger equation with polynomial potential

A functional integral representation for the weak solution of the Schrödinger equation with a polynomially growing potential is proposed in terms of an analytically continued Wiener integral. The asymptotic expansion in powers of the coupling constant λ of the matrix elements of the Schrödinger group is studied and its Borel summability is proved.     

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,α_*), where K=[0,+∞[×R and α_* a convolution product on K coming from the product formula satisfied by the Laguerre functions (m∈N, α?0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μα_*k=μα_*?α_*μ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.     

Edgeworth expansions of stochastic trading time

Under most local and stochastic volatility models the underlying forward is assumed to be a positive function of a time-changed Brownian motion. It relates nicely the implied volatility smile to the so-called activity rate in the market. Following Young and DeWitt-Morette (1986) [8], we propose to apply the Duru-Kleinert process-cum-time transformation in path integral to formulate the transition density of the forward. The method leads to asymptotic expansions of the transition density around a Gaussian kernel corresponding to the average activity in the market conditional on the forward value. The approximation is numerically illustrated for pricing vanilla options under the CEV model and the popular normal SABR model. The asymptotics can also be used for Monte Carlo simulations or backward integration schemes.     

A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

We evaluate a 1‐loop, 2‐point, massless Feynman integral I_D,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of I_D,m(p,q) in terms of powers of the variable X:=4p²/q⁴, where p=| p |, q=| q |, , , and in terms of generalised hypergeometric functions ₃F₂(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X^1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.     

Series transformations and integrals involving the Riemann Ξ-function

The transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar are unified, in the sense that all these formulas come from the same source, namely, a general formula involving an integral of Riemann's Ξ-function. We give proofs of all of these transformation formulas using the theory of Mellin transforms and the residue theorem. Our study includes new extensions of the formulas of Koshliakov and Ferrar through their connection with integrals involving the Riemann Ξ-function.     

Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO

In this paper we study boundedness properties of certain harmonic analysis operators (maximal operators for heat and Poisson semigroups, Riesz transforms and Littlewood-Paley g-functions) associated with Bessel operators, on the space BMOo(R) that consists of the odd functions with bounded mean oscillation on R.     

Numerical solution of the inverse spectral problem for Bessel operators

We consider some inverse spectral problems associated with the singular Sturm-Liouville equation

     

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions,for 1 and 2 unstable modes

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the x-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearance, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.