Papers by Mikhail Neklyudov
In this paper we will show that the solution of 1D stochastic heat equation with additive noise c... more In this paper we will show that the solution of 1D stochastic heat equation with additive noise converges to a martingale (independent upon space variable) when we "switch off" noise at the extremum points of the process. Contents 1. Introduction 1 2. Definitions 1 3. A priori estimates 3 4. Main result 9 5. Proofs of Proposition 4.1 and Theorem 4.1 10 References 12
We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cel... more We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cell-cell adhesive interactions. We rigorously address two PDE-based models, one featuring non-local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlaeger (1985).
A linear stochastic vector advection equation is considered; the equation may model a passive mag... more A linear stochastic vector advection equation is considered; the equation may model a passive magnetic field in a random fluid. When the driving velocity field is rough but deterministic, in particular just Hölder continuous and bounded, one can construct examples of infinite stretching of the passive field, arising from smooth initial conditions. The purpose of the paper is to prove that infinite stretching is prevented if the driving velocity field contains in addition a white noise component.
We show how the problems related to the Collatz map T can be transferred to the language of funct... more We show how the problems related to the Collatz map T can be transferred to the language of functional analysis. We associate with T certain linear operator 𝒯 and show that cycles and (hypothetical) diverging trajectory (generated by T) correspond to certain classes of fixed points of operator 𝒯. Furthermore, we demonstrate connection between dynamical properties of operator 𝒯 and map T. We prove that absence of nontrivial cycles of T leads to hypercyclicity of operator 𝒯. In the second part we construct an invariant measure for 𝒯 in a slightly different setup, deduce functional equation for characteristic function of total stopping time and investigate how the operator acts on generalized arithmetic progressions.
In this note we define one more way of quantization of classical systems. The quantization we con... more In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra 𝒪_∞ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie bracket and linearity.
ABSTRACT. The aim of this article is to study the asymptotic behaviour for large times of solutio... more ABSTRACT. The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in [14] for deterministic PDEs. CONTENTS
ABSTRACT. We consider a multidimensional Burgers equation in the whole space and show that there ... more ABSTRACT. We consider a multidimensional Burgers equation in the whole space and show that there exists a unique global solution if a Beale-Kato-Majda type condition is satisfied. In particular, if the initial condition and force have gradient form then we prove the global existence and uniqueness of solution and establish a new priori estimate. We will consider the following Burgers equation in Rd: ∂ui ∂t + n∑ j ∂ui u = ν△u ∂xj i + f i (0.1)
In this thesis, certain systems of linear parabolic equations called vector advection equations w... more In this thesis, certain systems of linear parabolic equations called vector advection equations will be considered. These equations are of great current scientific interest because they appear in magnetohydrodynamics and also it models certain properties of three dimensional Navier-Stokes equations which does not appear in the model of scalar advection. The thesis consists of six chapters. The first chapter is a review of existing relevant literature. The second chapter contains preliminary material necessary for further chapters. In the third chapter it is shown that solution of vector advection equations is self-dual in a certain sense described in the thesis. It is established that the so called regularity space of vector transport operator changes with time reversal of velocity 1'. Also the classical result of Serrin-Prodi-Ladyzhenkaya on the existence of strong solution of Navier-Stokes equations is reproved. In the fourth chapter the Feynman-Kac type formulas for the vecto...
arXiv: Probability, 2016
We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cel... more We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cell-cell adhesive interactions. We rigorously address two PDE-based models, one featuring non-local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlaeger (1985).
arXiv: Functional Analysis, 2019
Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogona... more Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schrodinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.
In this note we define one more way of quantization of classical systems. The quantization we con... more In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie bracket and linearity.
Russian Journal of Mathematical Physics
ABSTRACT
Analysis and Numerics, 2013
In this article we prove existence of global solution for random vortex filament equation. Our wo... more In this article we prove existence of global solution for random vortex filament equation. Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli [4] showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo [6] proved the existence of a local solution for a general initial condition.
We study ergodic properties of stochastic geometric wave equations on a particular model with the... more We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering the space variable-independent solutions only. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and we obtain also results on attractivity towards an invariant measure. We also present a suitable numerical scheme for approximating the solutions subject to a sphere constraint.
Stochastic Processes and their Applications, 2013
The aim of this article is to study the asymptotic behaviour for large times of solutions to a ce... more The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in [11] for deterministic PDEs.
Nonlinearity, 2013
We prove the existence of a global solution for the filament equation with inital condition given... more We prove the existence of a global solution for the filament equation with inital condition given by a geometric rough path in the sense of Lyons [17]. Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli [5] showed the existence of a global solution for a smooth initial condition while Bessaih, Gubinelli, Russo [6] proved the existence of a local solution for a general initial condition given by a rough path.
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Papers by Mikhail Neklyudov