Equilibria in some families of Kolmogorov systems

Kolmogorov systems constitute a general model for the dynamics of biological species. In that sense, they are generalizations of the Lotka–Volterra systems. Here, some classical results on existence, uniqueness, and global attraction of local equilibria that hold in Lotka–Volterra systems are generalized to Kolmogorov systems.     

Convergence in variation of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary measures

We study convergence in variation of probability solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions. We obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels. In addition, we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data. Finally, we obtain sufficient conditions for solvability of nonlinear Fokker–Planck–Kolmogorov equations.     

Heavy-tailed fractional Pearson diffusions

We define heavy-tailed fractional reciprocal gamma and Fisher–Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher–Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher–Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation.     

A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

Affinity of the Arov Entropy

In this work we continue the study of historically the first version of dynamical entropy. This version was suggested in master’s thesis by D. Arov and went practically unnoticed. The main result of the paper is that the Arov entropy, like the Kolmogorov–Sinai entropy, has the affine property. This, in particular, allows constructing a variety of dynamical systems where the Arov entropy is not determined by the Kolmogorov–Sinai entropy.     

On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

Doklady Mathematics - In this paper we give a positive answer to the question about the possibility of existence of several probability solutions to the Fokker–Planck–Kolmogorov...     

Estimates of distances between transition probabilities of diffusions

We obtain upper bounds for the total variation and entropy distances between probability solutions to two Fokker–Planck–Kolmogorov equations with different diffusion matrices and drifts     

Degenerate Elliptic Equations and Nonuniqueness of Solutions to the Kolmogorov Equation

Doklady Mathematics - In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker–Planck–Kolmogorov...     

The Kolmogorov–Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.     

Integrability and continuity of solutions to Fokker–Planck–Kolmogorov equations

New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained.     

On the Superposition Principle for Fokker–Planck–Kolmogorov Equations

Doklady Mathematics - Abstract—A generalization of the superposition principle for probability solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation is given,...     

One Inequality of the Landau–Kolmogorov Type for Periodic Functions of Two Variables

Ukrainian Mathematical Journal - We obtain a new sharp inequality of the Landau–Kolmogorov type for a periodic function of two variables estimating the convolution of the best uniform...     

Convergence to Stationary Measures in Nonlinear Fokker–Planck–Kolmogorov Equations

The convergence of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions is studied. Broad sufficient conditions for convergence in variation with an exponential bound are obtained.     

On the Kolmogorov–Feller law for exchangeable random variables

We prove a Kolmogorov–Feller weak law of large numbers for exchangeable sequences, under a second order hypothesis on the truncated mixands.     

On the Relationship between Sharp Kolmogorov-Type Inequalities and Sharp Kolmogorov–Remez-Type Inequalities

Ukrainian Mathematical Journal - We establish a new theorem on the relationship between sharp constants in Kolmogorov-type inequalities and sharp constants in Kolmogorov–Remez-type...     

Kolmogorov Widths of the Nikol’skii–Besov Classes of Periodic Functions of Many Variables in the Space of Quasicontinuous Functions

Ukrainian Mathematical Journal - We obtain order estimates for the M-dimensional Kolmogorov widths of the Nikol’skii–Besov classes of periodic functions of many variables with...     

On Non-Uniqueness of Probability Solutions to the Two-Dimensional Stationary Fokker–Planck–Kolmogorov Equation

The problem of uniqueness of probability solutions to the two-dimensional stationary Fokker–Planck–Kolmogorov equation is considered. Under broad conditions, it is proved that the existence of two different probability solutions implies the existence of an infinite set of linearly independent probability solutions.     

Uniqueness for Solutions of Fokker–Planck Equations on Infinite Dimensional Spaces

We develop a general technique to prove uniqueness of solutions for Fokker–Planck equations on infinite dimensional spaces. We illustrate this method by implementing it for Fokker–Planck equations in Hilbert spaces with Kolmogorov operators with irregular coefficients and both non-degenerate or degenerate second order part.     

Ergodicity for the 3D stochastic Navier–Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u_m of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.     

Analogues of Classical Goodness-of-Fit Tests for Distribution Tails

Doklady Mathematics - We propose analogues of the classical Kolmogorov–Smirnov and omega-squared tests for goodness-of-fit testing of distribution tails. The consistency of the proposed tests...