Hashimoto transform for stochastic Landau-Lifshitz-Gilbert equation

We show that Hashimoto transformation is applicable to the one dimensional stochastic Landau-Lifshitz-Gilbert (LLG) equation and transforms it to the stochastic generalized heat equation with nonlocal (in space) interaction.Comment: 9 page

Relationship between stochastic flows and connection forms

In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold.Comment: 6 page

Chernoff and Trotter-Kato theorems for locally convex spaces

We develop new approach for studying the abstract Cauchy problem $\dot{x}=Ax$, $x(0)=x_0\in D(A)$ for linear operators $A$ defined on a locally convex space $X$. This approach was firstly introduced in the paper "Chernoff and Trotter type product formulas" to study the problem for Banach spaces. In this paper we not only generalize the results of the previous paper to more general topological spaces but also get new results for Banach spaces. In particular, we prove the "local" extension of Chernoff-Trotter-Kato type theorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Kato theorems for locally convex spaces. Also we find necessary and sufficient conditions for the validity of the Chernoff and Trotter product formulas.Comment: 42 page

A particle system approach to cell-cell adhesion models

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).Comment: 22 page

Duality, Vector advection and the Navier-Stokes equations

In this article we show that three dimensional vector advection equation is self dual in certain sense defined below. As a consequence, we infer classical result of Serrin of existence of strong solution of Navier-Stokes equation. Also we deduce Feynman-Kac type formula for solution of the vector advection equation and show that the formula is not unique i.e. there exist flows which differ from standard flow along which vorticity is conserved.Comment: 51 pages; Some minor mistakes are correcte

Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs

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 \cite{[Ghidaglia-1986]} for deterministic PDEs.Comment: 31 pages;Major Changes: New theorem which covers the case of SPDEs with quadratic nonlinearity (such as stochastic Navier-Stokes equation with multiplicative noise) is adde

Global evolution of random vortex filament equation

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 (1998).Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli (2007) showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo (2005) proved the existence of a local solution for a general initial condition given by a rough path.Comment: 19 pages; minor revision

Ergodicity for infinite particle systems with locally conserved quantities

We analyse certain degenerate infinite dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a certain model of heat conduction. In particular, we establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to equilibrium is polynomial. Consequently, there is no spectral gap, but a Liggett-Nash type inequality is shown to hold.Comment: 34 pages; introduction rewritten, minor corrections, references adde

Distribution of small dispersive coal dust particles and absorbed radioactive chemical elements in conditions of forced acoustic resonance in iodine air filter at nuclear power plant

The physical features of distribution of the small dispersive coal dust particles and the adsorbed radioactive chemical elements and their isotopes in the absorber with the granular filtering medium with the cylindrical coal granules were researched in the case of the intensive air dust aerosol stream flow through the iodine air filter (IAF). It was shown that, at the certain aerodynamic conditions in the IAF, the generation of the acoustic oscillations is possible. It was found that the acoustic oscillations generation results in an appearance of the standing acoustic waves of the air pressure (density) in the IAF. In the case of the intensive blow of the air dust aerosol, it was demonstrated that the standing acoustic waves have some strong influences on both: 1) the dynamics of small dispersive coal dust particles movement and their accumulation in the IAF; 2) the oversaturation of the cylindrical coal granules by the adsorbed radioactive chemical elements and their isotopes in the regions, where the antinodes of the acoustic waves are positioned. Finally, we completed the comparative analysis of the theoretical calculations with the experimental results, obtained for the cases of: 1) the experimental aerodynamic modeling of physical processes of the absorbed radioactive chemical elements and their isotopes distribution in the IAF; and 2) the gamma-activation spectroscopy analysis of the absorbed radioactive chemical elements and their isotopes distribution in the IAF. We made the innovative propositions on the necessary technical modifications with the purpose to improve the IAF technical characteristics and increase its operational time at the nuclear power plant (NPP), going from the completed precise characterization of the IAF parameters at the long term operation.Comment: 8 pages, 2 figures, http://vant.kipt.kharkov.ua/ARTICLE/VANT_2013_2/article_2013_2_94.pd

Chernoff and Trotter type product formulas

We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the solution x(\cdot) can be represented as a limit of sequence F(t/n)^{n} as n\to\infty in the weak operator topology, where a function F:[0,\infty)\to L(X) satisfies F'(0)y=Ay, y\in D(A). As a consequence, we deduce necessary and sufficient conditions that a linear operator C is closable and its closure is a generator of C_0-semigroup. We also obtain some criteria for the sum of two generators of C_0-semigroups to be a generator of C_0-semigroup such that the Trotter formula is valid.Comment: 20 page