85 results found
Buttà P, Flandoli F, Ottobre M, et al., 2019, A non-linear kinetic model of self-propelled particles with multiple equilibria, Kinetic and Related Models, Vol: 12, Pages: 791-827, ISSN: 1937-5093
We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density ft, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only hypoelliptic) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate N-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as N→∞, to the solution of the PDE.
Liu X, Zegarliński B, 2018, On Continuous Coding, International Conference on Stochastic Partial Differential Equations and Related Fields, Pages: 539-547, ISSN: 2194-1017
Zegarlinski B, 2017, Crystallographic Groups for Hormander Fields, Science Journal of Volgograd State University. Mathematics. Physics, Vol: 20, Pages: 43-64, ISSN: 2222-8896
This is a preview paper on Crystallographic Groups of Hörmander Fields. We describe an emerging picture in analysis of extended groups. In particular, we introduce and provide examples of Crystallographic Groups associated to a Hörmander system of fields as well as discuss some related analysis.
Kontis V, Ottobre M, Zegarlinski B, 2017, Long- and short-time behaviour of hypocoercive-type operators in infinite dimensions: an analytic approach, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol: 30, ISSN: 0219-0257
In this paper we provide a range of examples to illustrate the general theory developed in Ref. 19, where we studied smoothing and ergodicity for infinite dimensional Markovian systems with hypocoercive type generator. We also introduce and study new models, where the framework of Ref. 19 cannot be applied as is but can be adapted to obtain improved results, by exploiting the specific structure of the generator at hand. Among such examples, we examine a system of infinitely many interacting heat baths.
Pierre Fougers, Ivan Gentil, Zegarlinski B, 2017, Solution of a class of reaction-diffusion systems via logarithmic Sobolev inequality, Annales Mathématiques Blaise Pascal, Vol: 24, Pages: 1-53
We study global existence, uniqueness and positivity of weak solutions of a class of reaction-diffusion systems coming from chemical reactions. The principal result is based only on a logarithmic Sobolev inequality and the exponential integrability of the initial data. In particular we develop a strategy independent of dimensions in an unbounded domain.
Zegarlinski B, 2016, Linear and nonlinear dissipative dynamics, Reports on Mathematical Physics, Vol: 77, Pages: 377-397, ISSN: 1879-0674
In this paper we introduce and study new dissipative dynamics forlarge interacting systems.
V Kontis, MOttobre, Zegarlinski B, 2016, Markov semigroups with hypocoercive-type generator in infinite dimensions: Ergodicity and smoothing, Journal of Functional Analysis, Vol: 270, Pages: 3173-3223, ISSN: 0022-1236
We start by considering finite dimensional Markovian dynamics in Rm generated by operators of hypocoercive type and for such models we obtain short and long time pointwise estimates for all the derivatives, of any order and in any direction, along the semigroup. We then look at infinite dimensional models (in (Rm)Zd) produced by the interaction of infinitely many finite dimensional dissipative dynamics of the type indicated above. For these infinite dimensional models we study finite speed of propagation of information, well-posedness of the semigroup, time behaviour of the derivatives and strong ergodicity problem.
Al-Rashed MHA, Zegarlinski B, 2014, Monotone norms and Finsler structures in noncommutative spaces, INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, Vol: 17, ISSN: 0219-0257
Lugiewicz P, Olkiewicz R, Zegarlinski B, 2013, NONLINEAR MARKOV SEMIGROUPS ON C*-ALGEBRAS, INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, Vol: 16, ISSN: 0219-0257
Inglis J, Neklyudov M, Zegarlinski B, 2012, ERGODICITY FOR INFINITE PARTICLE SYSTEMS WITH LOCALLY CONSERVED QUANTITIES, INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, Vol: 15, ISSN: 0219-0257
Pierre Fougeres, Cyril Roberto, Boguslaw Zegarlin ski, 2012, Sub-gaussian measures and associated semilinear problems, Revista Mathematica Iberoamericana, Vol: 28, Pages: 305-350, ISSN: 0213-2230
Al-Rashed MHA, Zegarlinski B, 2011, Noncommutative Orlicz spaces associated to a state II, LINEAR ALGEBRA AND ITS APPLICATIONS, Vol: 435, Pages: 2999-3013, ISSN: 0024-3795
Brzezniak Z, Flandoli F, Neklyudov M, et al., 2011, Conservative Interacting Particles System with Anomalous Rate of Ergodicity, JOURNAL OF STATISTICAL PHYSICS, Vol: 144, Pages: 1171-1185, ISSN: 0022-4715
Inglis J, Kontis V, Zegarlinski B, 2011, From U-bounds to isoperimetry with applications to H-type groups, Journal of Functional Analysis (2011), Vol: 260, Pages: 76-116
Dragoni F, Kontis V, Zegarlinski B, 2011, Ergodicity of Markov Semigroups with Hörmander Type Generators in Infinite Dimensions, Potential Analysis: an international journal devoted to the interactions between potential theory, p
Zegarlinski B, 2011, Analysis on Extended Heisenberg Group, Annales de la faculté des sciences de Toulouse Sér. 6, Vol: 20, Pages: 379-405
Xu L, Zegarlinski B, 2010, Existence and Exponential mixing of infinite white alpha-stable Systems with unbounded interactions, ELECTRONIC JOURNAL OF PROBABILITY, Vol: 15, Pages: 1994-2018, ISSN: 1083-6489
Lugiewicz P, Olkiewicz R, Zegarlinski B, 2010, Ergodic properties of diffusion-type quantum dynamical semigroups, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 43, ISSN: 1751-8113
Gentil I, Zegarlinski B, 2010, ASYMPTOTIC BEHAVIOUR OF REVERSIBLE CHEMICAL REACTION-DIFFUSION EQUATIONS, KINETIC AND RELATED MODELS, Vol: 3, Pages: 427-444, ISSN: 1937-5093
Hebisch W, Zegarlinski B, 2010, Coercive inequalities on metric measure spaces, Journal of Functional Analysis 258 (2010) 814–851
Bobkov S, Zegarlinski B, 2010, Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions, Around the Research of Vladimir Maz'ya I, Editors: Laptev, Publisher: Springer, Pages: 13-79, ISBN: 978-1-4419-1340-1
Xu L, Zegarlinski B, 2009, Ergodicity of the Finite and Infinite Dimensional -Stable Systems, STOCHASTIC ANALYSIS AND APPLICATIONS, Vol: 27, Pages: 797-824, ISSN: 0736-2994
Olkiewicz R, Xu L, Zegarlinski B, 2008, Nonlinear problems in infinite interacting particle systems, INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, Vol: 11, Pages: 179-211, ISSN: 0219-0257
Zegarlinski B, 2008, Nonlinear Markovian problems in large dimensions, 5th Seminar on Stochastic Analysis, Random Fields and Applications, Publisher: BIRKHAUSER BOSTON, Pages: 397-408
Zegarlinski B, 2008, Analysis in operator spaces, Quantum Stochastics and Information: Statistics, Filtering, and Control, Pages: 121-140, ISBN: 9789812832955
We briey review a number of issues of analysis in operator spaces. We begin from introducing and recalling few results (and open problems) on ergodicity of dissipative dynamics in inductive limit C algebra. As a mo- tivation we present later some recent achievements in commutative case saying that one can obtain stronger ergodicity results by analysis based on a notion of hypercontractivity of dissipative dynamics in suitable functional (Lp and Orlicz) spaces. The hypercontractivity property, besides helping in study the ergodicity, has an intricate relation to the entropy and the parti- cle structure of a system which provides additional motivation to research in this area.
P Ługiewicz, Zegarlinski B, 2007, Coercive inequalities for Hörmander type generators in infinite dimensions, Journal of Functional Analysis, Vol: 247 (2007) 438–476, Pages: 438-476, ISSN: 0022-1236
Zegarlinski B, 2007, Linear and nonlinear phenomena in large interacting systems, REPORTS ON MATHEMATICAL PHYSICS, Vol: 59, Pages: 409-419, ISSN: 0034-4877
Roberto C, Zegarlinski B, 2007, Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups, Journal of Functional Analysis, Vol: 243, Pages: 28-66, ISSN: 0022-1236
Al-Rashed MHA, Zegarlinski B, 2007, Noncommutative Orlicz spaces associated to a state, STUDIA MATHEMATICA, Vol: 180, Pages: 199-209, ISSN: 0039-3223
Albeverio S, Liang S, Zegarlinski B, 2006, Remark on the integration by parts formula for the phi(4)(3)-quantum field model, INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, Vol: 9, Pages: 149-154, ISSN: 0219-0257
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