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We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

We present an algorithm that reveals relevant contributions in non-threshold-type asymptotic expansion of Feynman integrals about a small parameter. It is shown that the problem reduces to finding a convex hull of a set of points in a multidimensional vector space.

This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.

This work, which can be considered as a small monograph, is devoted to the study of two-and three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of compatibility of asymptotic expansions, the boundary layer method, and the multiscale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very essential weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

Neurosecretory cells spatially redistribute their pool of secretory vesicles upon stimulation. Recent observations suggest that in chromaffin cells vesicles move either freely or in a directed fashion by what appears to be a conveyor belt mechanism. We suggest that this observation reflects the transient active transport through molecular motors along cytoskeleton fibres and quantify this effect using a 1D mathematical model that couples a diffusion equation to advection equations. In agreement with recent observations the model predicts that random motion dominates towards the cell centre whereas directed motion prevails in the region abutting the cortical membrane. Furthermore the model explains the observed bias of directed transport towards the periphery upon stimulation. Our model suggests that even if vesicle transport is indifferent with respect to direction, stimulation creates a gradient of free vesicles at first and this triggers the bias of transport in forward direction. Using matched asymptotic expansion we derive an approximate drift-diffusion type model that is capable of quantifying this effect. Based on this model we compute the characteristic time for the system to adapt to stimulation and we identify a Michaelis–Menten-type law describing the flux of vesicles entering the pathway to exocytosis.

The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R^sup 3^ that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The three-term asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers x1, . . . , xN of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N » 1, is also derived. [PUBLICATION ABSTRACT]

The asymptotic behavior of the analytic solutions of a family of singularly perturbed q -difference–differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to the time variable are provided: one of Gevrey nature, and another of mixed type Gevrey and q -Gevrey. These asymptotic phenomena are observed due to the modification of the norm established on the space of coefficients of the formal solution. The techniques used are based on the adequate path deformation of the difference of two analytic solutions, and the application of several versions of Ramis–Sibuya theorem.

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results.

In this paper we consider the multiscale analysis of a Steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure. There are mainly two new results in the present paper. First, we obtain the convergence rate with ε ½ for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the Steklov eigenvalue problem. Second, the boundary layer solution is defined. Numerical simulations are then carried out to validate the above theoretical results.

Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show in this paper a bridge representation for the joint density of the lognormal fractional SABR model in a Fourier space. Evaluating the bridge representation along a properly chosen deterministic path yields a small time asymptotic expansion to the leading order for the probability density of the fractional SABR model. A direct generalization of the representation of joint density often leads to a heuristic derivation of the large deviations principle for joint density in a small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients.