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Sergei Kuksin

Publications and source records attributed to Sergei Kuksin.

At least 19 recordsLinked to original sources
arXiv2026

Long-time behaviour of dynamical systems driven by bounded mixing noises

We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are reduced to discrete-time random dynamical systems generated by time-one maps, so that the main analysis is carried out in the discrete setting. We introduce a class of mixing random forcings whose regular conditional distributions with respect to the past satisfy natural regularity, recurrence, and non-degeneracy assumptions, extending the framework previously developed for more restrictive classes of processes in a paper by Kuksin-Shirikyan in GAFA (2025). Under a linearised controllability assumptions on the system, we prove exponential mixing in the total variation metric for finite-dimensional phase spaces. We then establish an infinite-dimensional counterpart yielding exponential mixing in the dual-Lipschitz metric under suitable amendments of restrictions on the system and the random forcing. Our approach is based on lifting the dynamics to an appropriate Markov process on an infinite-dimensional history space and applying a Doeblin coupling argument through the method of Kantorovich functional. As applications, we derive exponential mixing for a broad class of ordinary differential equations driven by mixing random processes with bounded continuous trajectories. As an application of our result to PDEs we discuss the randomly perturbed primitive equations of atmospheric dynamics.

math.DS↗
arXiv2025

Mixing for dynamical systems driven by stationary noises

The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be $δ$-correlated in time, so that the evolution in question is not necessarily Markovian. We first prove an abstract result which imply the mixing for random dynamical systems satisfying appropriate dissipativity and controllability conditions. It is applicable to a large class of evolution equations, and we illustrate it on the examples of a chain of anharmonic oscillators coupled to heat reservoirs, the 2d Navier-Stokes system, and a complex Ginzburg-Landau equation. Our results also apply to the general theory of random processes on the 1d lattice and allow one to get for them results related to Dobrushin's theorems on reconstructing processes via their conditional distributions. The proof is based on an iterative construction with quadratic convergence. It uses the method of Kantorovich functional, introduced in [KPS02, Kuk02, Kuk06] in the context of randomly forced PDEs, and some ideas suggested in [Shi15, KNS20] to prove mixing with the help of controllability properties of an associated system

math.PR↗
arXiv2025

Markovian reduction and exponential mixing in total variation for random dynamical systems

The paper deals with the problem of large-time behaviour of trajectories for discrete-time dynamical systems driven by a random noise. Assuming that the phase space is finite-dimensional and compact, and the noise is a Markov process with a transition probability satisfying some regularity hypotheses, we prove that all the trajectories converge to a unique measure in the total variation metric. The proof is based on the Markovian reduction of the system in question and a result on mixing for Markov processes. Then we present an extension of this result to the case of systems driven by stationary noises.

math.PR↗
arXiv2025

On the averaging theorems for stochastic perturbation of conservative linear systems

For stochastic perturbations of linear systems with non-zero pure imaginary spectrum we discuss the averaging theorems in terms of the slow-fast action-angle variables and in the sense of Krylov-Bogoliubov. Then we show that if the diffusion matrix of the perturbation is uniformly elliptic, then in all cases the averaged dynamics does not depend on a hamiltonian part of the perturbation.

math.DS↗
arXiv2024

Averaging for stochastic perturbations of integrable systems

We are concerned with averaging theorems for $ε$-small stochastic perturbations of integrable equations in $\mathbb{R}^d \times \mathbb{T}^n =\{(I,φ)\}$ $$ \dot I(t) =0,\quad \dot φ(t) = θ(I), \qquad (1)$$ and in $\mathbb{R}^{2n} = \{v=(\mathbf{v}_1, \dots, \mathbf{v}_n), \; \mathbf{v}_j \in \mathbb{R}^2\}$, $$ \dot{\mathbf{v}}_k(t) =W_k(I) \mathbf{v}_k^\bot, \quad k=1, \dots, n, \qquad (2) $$ where $I=(I_1, \dots, I_n)$ is the vector of actions, $I_j = \frac12 \| \mathbf{v}_j\|^2$. The vector-functions $θ$ and $W$ are locally Lipschitz and non-degenerate. Perturbations of these equations are assumed to be locally Lipschitz and such that some few first moments of the norms of their solutions are bounded uniformly in $ε$, for $0\le t\le ε^{-1} T$. For $I$-components of solutions for perturbations of (1) we establish their convergence in law to solutions of the corresponding averaged $I$-equations, when $0\le τ:= εt\le T$ and $ε\to0$. Then we show that if the system of averaged $I$-equations is mixing, then the convergence is uniform in the slow time $τ=εt\ge0$. Next using these results, for $ε$-perturbed equations of (2) we construct well posed {\it effective stochastic equations} for $v(τ)\in \mathbb{R}^{2n}$ (independent from $ε$) such that when $ε\to0$, actions of solutions of the perturbed equations of (2) with $t:= τ/ε$ converge in distribution to actions of solutions for the effective equations. Again, if the effective system is mixing, this convergence is uniform in the slow time $τ\ge0$. We provide easy sufficient conditions on the perturbed equations which ensure that our results apply to their solutions.

math.PR↗
arXiv2024

Stochastic 1d Burgers equation as a model for hydrodynamical turbulence

This work is a review with proofs of a group of results on the stochastic Burgers equation with small viscosity, obtained during the last two decades. These results jointly show that the equation makes a surprisingly good model of hydrodynamical turbulence. The model provides natural and rigorously justified analogies of a number of key predictions of the theory of turbulence, including the main assertions of the Kolmogorov approach to turbulence, known as the K41 theory.

math-ph↗
arXiv2023

Averaging and mixing for stochastic perturbations of linear conservative systems

We study stochastic perturbations of linear systems of the form $$ dv(t)+Av(t)dt = εP(v(t))dt+\sqrtεB(v(t)) dW (t), v\in\mathbb{R}^{D}, (*) $$ where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field $P(v)$ and the matrix-function $B(v)$ are locally Lipschitz with at most a polynomial growth at infinity, that the equation is well posed and first few moments of norms of solutions $v(t)$ are bounded uniformly in $ε$. We use the Khasminski approach to stochastic averaging to show that as $ε\to0$, a solution $v(t)$, written in the interaction representation in terms of operator $A$, for $0\le t \le Const\,ε^{-1}$ converges in distribution to a solution of an effective equation. The latter is obtained from (*) by means of certain averaging. Assuming that eq.(*) and/or the effective equation are mixing, we examine this convergence further.

math.DS↗
arXiv2023

Weak and strong versions of the Kolmogorov 4/5-law for stochastic Burgers equation

For solutions of the space-periodic stochastic 1d Burgers equation we establish two versions of the Kolmogorov 4/5-law which provides an asymptotic expansion for the third moment of increments of turbulent velocity fields. We also prove for this equation an analogy of the Landau objection to possible universality of Kolmogorov's theory of turbulence, and show that the third moment is the only one which admits a universal asymptotic expansion.

math.PR↗
arXiv2023

A refinement of Heath-Brown's theorem on quadratic forms

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be $C_0^\infty$-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not vanish at the singularity and have an explicit decay at infinity. The paper uses only elementary results from the number theory and is available to readers without a number-theoretical background.

math.NT↗
arXiv2022

Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition (complete version)

In this paper we continue to study small amplitude solutions of the damped cubic NLS equation, driven by a random force (the study was initiated in our previous work [A.Dymov, S.Kuksin, Comm. Math. Phys.'2021] and continued in [A.Dymov, S.Kuksin, A.Maiocchi, S.Vladuts, arXiv:2104.11967]). We write solutions of the equation as formal series in the amplitude and discuss the behaviour of this series under the wave turbulence limit, when the amplitude goes to zero, while the space-period goes to infinity.

math-ph↗
arXiv2022

On averaging and mixing for stochastic PDEs

We examine the convergence in the Krylov--Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.

math.PR↗
arXiv2021

The large-period limit for equations of discrete turbulence

We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $λ$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first $λ\to 0$ and then $L\to \infty$. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit $L\to\infty$ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.

math.AP↗
arXiv2021

Some remarks on Heath-Brown's theorem on quadratic forms

In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to be $C_0^\infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.

math.NT↗
arXiv2021

Kolmogorov's theory of turbulence and its rigorous 1d model

This paper is a synopsis of the recent book A. Boritchev, S. Kuksin, \textit{One-Dimensional Turbulence and the Stochastic Burgers Equation}, AMS Publications, 2021 (to appear). The book is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.

math-ph↗
arXiv2020

Krylov--Bogolyubov averaging

We present the modified approach to the classical Bogolyubov-Krylov averaging, developed recently for the purpose of PDEs. It allows to treat Lipschitz perturbations of linear systems with pure imaginary spectrum and may be generalized to treat PDEs with small nonlinearities.

math.DS↗
arXiv2020

Formal expansions in stochastic model for wave turbulence 1: kinetic limit

We consider the damped/driver (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove that when the amplitude goes to zero and the torus' size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.

math-ph↗
arXiv2020

On The Energy Transfer To High Frequencies In The Damped/Driven Nonlinear Schrödinger Equation (Extended Version)

We consider a damped/driven nonlinear Schrödinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-νΔu+i|u|^2u=\sqrtνη(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad ν>0, \] where $η(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le Cν^{-m}, $ uniformly in $t\ge0$ and $ν>0$. In this work we prove that for small $ν>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ become large at least to the order of $ν^{-κ_{n,m}}$ with $κ_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}ν)$.

math.AP↗